Even in the elite world of private banking, Bessemer Trust Co. stands apart.

Founded in 1907 to manage a steel magnate's fortune, Bessemer has evolved into a full-fledged bank, albeit one that caters to an exceptionally wealthy clientele. Clients need $5 million just to get in the door, and their accounts average $10 million.

Of 2,500 so-called family offices currently in existence, only about a dozen have successfully attracted new customers to their fold. And Bessemer, which converted to a bank charter in 1974, is the largest of this rare breed, with $9 billion of assets under management.

These days, members of the founding Phipps family account for only about a fifth of Bessemer's business. But the attributes they sought in a financial adviser - discreet, personal service - are still much in evidence.

"Most people don't like to talk about their money in public, and we understand that," said John R. Whitmore, Bessemer's president and chief executive, in an interview in his midtown Manhattan office.

An imposing man with a thundering voice, Mr. Whitmore, 61, is credited with having built Bessemer into a modern private bank.

Since he took the helm in 1975, assets have grown ninefold from the Phipps' $1 billion, and the client roster has expanded to 600 individuals and 100 companies and endowments.

Though its parent company, the Bessemer Group, is based in Woodbridge, N.J., Bessemer Trust conducts most of its activities from a suite of wood- paneled offices on New York's Fifth Avenue.

The trust company never talks about its clients. But some names have come to light, courtesy of federal laws requiring top elected officials and political appointees to disclose their financial arrangements. Among them: George and Barbara Bush and former Treasury secretaries Donald Regan, Nicholas Brady, and Lloyd Bentsen.

Observers give Bessemer high marks for establishing itself in a market where reputation is everything, and for attracting exceptional talent.

Still, they wonder whether Bessemer's growth can continue unabated. These observers argue that competition is rising, fewer fortunes are being minted in the 1990s, the company has lost some key executives, and costly investments in technology could sap profits.

They also maintain that the Phipps family retains too much influence over Bessemer Trust, given that 80% of fee income now comes from nonfamily clients. Recently, Stuart S. Janney 3d, a Phipps cousin, succeeded Ogden "Danny" Phipps as chairman.

"What's good for the Phippses is what's good for everybody else. But that just doesn't work in today's market anymore," said a former Bessemer executive.

Mr. Whitmore knows what the critics are saying - and he will have none of it. He expects growth to continue at a brisk 10% a year, with the firm taking on no more than 70 new clients annually.

Indeed, he said, Bessemer recently entered discussions with a Santa Barbara, Calif., family with $75 million of assets and another prospective client who is selling a business for more than $100 million.

As for staff turnover, Mr. Whitmore attributed several departures to retirement. Others, he said, were courted by rival firms, a tribute to Bessemer's cachet.

He maintained that the company's systems are perfectly up to the challenge of handling new business. Indeed, he said, the company decided to seek only 70 new clients a year precisely because it doesn't want to scrimp on service.

"You never want to offend an old friend to make a new friend," Mr. Whitmore said in an obliging manner. "You want to keep the old friends happy and add a few."

And what of the Phippses?

Mr. Whitmore credits them with bringing in some of the new clients and assuring them that they would not be second-class citizens. "One of the elements of success has been the cooperation and support of the Phipps family," he said.

And despite a recent round of rumors that the Phipps family would place Bessemer up for sale, Mr. Whitmore said during a June telephone interview that he had been advised "they have no intention of selling the company to anyone."

For the first 68 years of its existence, Bessemer catered exclusively to descendents of Henry Phipps, a onetime partner in Carnegie Steel.

"The office," as the Phipps family calls Bessemer Trust to this day, handled every aspect of their personal financial affairs: money management, tax planning, travel reservations, even the financial side of running a stable of race horses. (The Phippses remain one of the premier families in horse racing.)

But by the mid-1970s, it became clear that the cost of these exclusive services was too great even for a family whose wealth had blossomed to $1 billion.

So Bessemer applied for and got a banking charter, allowing it to fully open its doors to nonfamily clients.

At the same time, Mr. Whitmore was brought in from Washington, where he had been a senior vice president at American Security and Trust, now a part of NationsBank Corp.

The company he helped build certainly has its share of admirers - even among its rivals.

"For a small organization, they are extremely well run," says Mary B. Lehman, Bankers Trust private bank managing director. "Their name comes up a lot."

Indeed, Bessemer Trust is on something of expansion drive. The company currently has offices in New York, New Jersey, Washington, Chicago, Miami, Palm Beach, London, and the Cayman Islands. Now, Mr. Whitmore said, it is opening a Naples, Fla., office, has bought a 5% share in the Commerce Bank of Seattle to check out the area, and is eyeing California for a possible office.

The company has a very focused view of what it wants to do, rivals say.

"They have concentrated on management and trust; they're not out there making commercial loans," said William L. Gunlicks, chairman of Founders National Trust Bank, Naples, Fla. "They're truly a private bank."

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