In 1957, the Tropicana opened as the most lavish hotel and casino in Las Vegas, with a cascading 60-foot fountain and shimmering pool that piped Muzak underwater. The magicians Siegfried and Roy debuted there; Sean Connery’s James Bond stayed the night; and the Corleones of The Godfather took over a fictionalized version of the resort. Feathered showgirls danced in its Folies Bergère cabaret, sealing the Tropicana’s reputation as synonymous with the Strip’s swinging midcentury glory.

But on a recent weekday afternoon, just a few patrons played flashing slot machines inside the dimly lit, smoky casino. Beneath its ornate stained-glass ceiling, some of the blackjack and roulette tables were closed. A handful of people drank at the bar, where they watched the Vegas Golden Knights, the city’s new N.H.L. team. Outside, in the near 100-degree heat, middle-aged visitors in bikinis languished near an artificial rock formation by the pool.

Now, decades since the heyday of the Tropicana, Las Vegas is offering a form of entertainment adopted by other major American cities: professional sports. The faded resort, one of the city’s oldest casinos, stands in the way on lucrative land.

Last week, the governor of Nevada signed a bill agreeing to finance up to $380 million of the $1.5 billion stadium to lure the Oakland Athletics to a site on the Las Vegas Strip. The project, which has yet to be finalized, could mean the demolition of the Tropicana Las Vegas, as it is now known. The resort would become another casualty in a city that is ever-changing to please its visitors; one notorious for imploding the old to make way for the new.

Las Vegas was once considered taboo for major sports because of its strong association with gambling on games, which is now legal in many states. If the A’s move to the city, the Major League Baseball team would follow the N.F.L., the N.H.L., the W.N.B.A. and the N.C.A.A. men’s basketball tournament. Last week, the Golden Knights, the city’s first major sports franchise, won the Stanley Cup.

While some welcome the city’s ascendance as a pro sports destination, others are nostalgic for a bygone era embodied by the Tropicana — when lounge acts like the Frank Sinatra-led Rat Pack made Vegas world famous and sparkling resorts materialized in the desert. Now, all but a handful of the casinos from that era are gone.

Often, their demolition has been made spectacle: In 1993, a crowd estimated at 200,000 people gazed on as a faux pirate ship at the newly-opened Treasure Island resort appeared to blow up The Dunes (built in 1955) to make way for the $1.6 billion Bellagio Hotel and Casino. Three years later, The Hacienda (1956) was wired with hundreds of New Year’s Eve fireworks that brought it tumbling down in a blazing display of light. In 2016, the Riviera Hotel and Casino (1955) was also serenaded by fireworks before being imploded into a pile of debris and dust.

“It’s sort of an inevitable kind of process here in Las Vegas, where we’re in a constant state of evolution,” said Geoff Schumacher, the author of “Sun, Sin & Suburbia: The History of Modern Las Vegas” and an executive at the city’s Mob Museum. But, he added, “that doesn’t mean we can’t be a little bit sad about it.”

On Thursday, Gov. Joe Lombardo of Nevada, a Republican, signed the bill approved by the Democratic Legislature to allow for the portion of public financing for the 30,000-seat stadium, inching the Tropicana closer to its demise. The next step in that plan will be the approval of M.L.B.’s owners to relocate — a process that Commissioner Rob Manfred said could come together quickly when he was asked about it last month. Bally’s Corporation, which owns the hotel and casino, had already reached a deal with the A’s to assign nine acres of the 35-acre parcel to the stadium.

In a memo sent in May to staff, the company said that operations at the Tropicana could cease within the next two years, but that the site would eventually reopen as a “new integrated casino hotel resort and baseball park.” Staff, it added, would lose their jobs. “We anticipate that employees will find an enhanced career opportunity when the casino hotel resort reopens,” the company said.

Soo Kim, the chairman of Bally’s, said in a text message that the memo had since been redacted and that many of the resort’s plans were still “up in the air.” A revised memo provided on Sunday by Mr. Kim did not specify a timeline for closure and said instead that the Tropicana would stay open “as long as practicable.”

In an interview, Mr. Kim said that the renovation was ultimately an opportunity to grow employment.

“The Trop is obviously iconic, but it is, really, in a lot of ways, economically obsolete,” Mr. Kim said, noting that the 35-acre site was worth more than the current economic capacity of the casino. “It literally is part of the glitz and glamour of Vegas, but it hasn’t been that for decades,” he added, noting that when it came to signing a deal to develop the site for a baseball stadium, it was a “no-brainer.”

The Culinary Workers Union Local 226, which represents tens of thousands of workers across Nevada, has vowed to protect employees of the Tropicana. “The unit can help them find work during the remodel or the rebuild, and these workers have to be hired back first,” Ted Pappageorge, the secretary-treasurer of the union, said in an interview, noting that the workers’ contract stipulates they get a severance package. Not all of the casino’s workers, however, are in the union. “They’re at the whim of the employer,” Mr. Pappageorge said.

Tawana Moore, who has been a guest room attendant at the Tropicana for 17 years, and is a union member, said on Thursday that workers had not heard anything from the company since the bill was signed into law. “We have made this our home,” Ms. Moore said in an earlier interview. “We’re sad, but happy at the same time — sad because we’ve been here forever, and some people don’t know how to start over, and some people are happy, because change is good. I welcome change.”

The looming stadium deal is not the first time the Tropicana has been threatened. Shortly after the 300-room resort opened, an attempt on the life of the crime boss Frank Costello exposed his ties to the casino, plunging it into controversy. By the 1970s, the resort was already struggling to compete with larger operations like Caesars Palace, and at the end of that decade, the F.B.I. exposed a mob-skimming operation that eventually forced the owners to sell the property. The Tropicana has since changed hands several times.

At every turn, the resort has tried to reinvent itself, morphing in the late 1980s into a family-friendly destination, complete with a water park, flamingoes and talking parrots — rebranded “The Island of Las Vegas.” Visitors could swim up to the blackjack table, and dry their money, according to an article in The New York Times. Twenty-five years later, the hotel was redone again, this time to mimic the atmosphere of South Beach, a luxury neighborhood in Miami, Fla. The tagline: “Surprise — it’s the new Tropicana Las Vegas.”

For nearly seven decades, the Tropicana has reflected “the changing images of the Strip,” said Stefan Al, an architect and the author of “The Strip: Las Vegas and the Architecture of the American Dream.”

“Maybe for the previous generation that was going to the casino and pressing a button on the slot machine. But for today’s generation, that may mean seeing a sports event, but also betting around that,” he said. “It’s a very different form of leisure.”

Some are lamenting the impending demise of another piece of the city’s history, but are not at all surprised. “I do miss old Vegas, but everything’s going to change eventually, so you might as well get used to it,” said Tyler Ricca, a tattooed man in cargo shorts who on a weekday afternoon, was playing four card keno and smoking Marlboros inside the Tropicana.

Mr. Ricca, 36, a regular, shrugged and looked around — the casino was largely empty, save for a few people who said they were escaping the crowds of the larger casinos. Outside, a hot breeze rippled through the fronds of towering palms.

“I was sad,” Kathi Thorpe, a Canadian visitor, said of the closure as she headed to the pool. But, she added, “It’s Vegas. You expect things to get torn down and rebuilt.”