As a thank-you to its most famous customer, Amtrak is renaming the train station in Wilmington, Del., after stimulus “sheriff” Vice President Joseph R. Biden — after the project received $20 million in stimulus money and came in $5.7 million over the initial announced budget.

Spokesmen for Mr. Biden, who said he personally fought for stimulus money for Amtrak, didn’t respond to messages Wednesday or Thursday.

But Amtrak said it alone made the decision to rename the station in honor of the vice president. The company also said its own budget projection was always greater than the $32 million the White House announced back in 2009 — though Amtrak itself also listed the $32 million cost that year, later raising the price tag to $37.7 million.

“The Wilmington station is owned by Amtrak, and Amtrak on its own made the determination to name the station,” said spokeswoman Danelle Hunter.

President Obama, soon after signing the Recovery Act in early 2009, designated Mr. Biden his “sheriff” overseeing the stimulus funds, charging him with regularly checking in with governors and mayors to try to prevent waste, fraud and abuse in the $821 billion economic recovery package.

In that role, Mr. Biden himself traveled to Wilmington on May 4, 2009, to announce the $20 million in stimulus money as part of what the White House billed a “$32 million renovation and restoration” of the century-old station.

“We would not be spending a penny on it today without the Recovery Act,” Mr. Biden said. “We have no time to wait. We cannot wait. That’s what the Recovery Act is about.”

The vice president, who as a senator commuted daily by train from his home in Delaware to Washington, bragged he had made more than 7,000 round trips on the track. He said he “heavily influenced the amount of money” that Amtrak received under the stimulus.

Renaming the station is apparently legal — and is fitting in this case, said Leslie Paige, spokeswoman for Citizens Against Government Waste.

“It’s an absolutely perfect monument to a guy whose entire history has been overspending and overpromising,” she said. “It would make sense [Mr. Biden‘s] name would be slapped on a bloated, over-budget train station in Wilmington.”

After more than 18 months of work, the station reopened in December with newly renovated real and faux-marble floors, a new heating and air-conditioning system, and free wireless Internet installed throughout. The renovation also improved the track’s waterproofing and reworked the station’s plumbing.

But the cost figures for the refurbishing project remain murky.

Not only did the White House cite the projected cost at $32 million, but Amtrak used that figure in a 2009 press release documenting its Delaware projects.

But Ms. Hunter, the Amtrak spokeswoman, said the company’s original budget for the renovation was actually $35.7 million: with $20 million from stimulus money; $12 million from Delaware’s Department of Transportation; and $3.7 million from Amtrak itself. Ms. Hunter said the final cost reached $37.7 million when Amtrak added $2 million worth of work that was not part of the original scope of the project.

In its official Recovery Act report, Amtrak gave a different figure altogether — $36 million to the dollar — for the refurbishing project. And it said the breakdown in funding was $20 million in stimulus funds and $4 million from a 2009 federal grant, in addition to the $12 million from Delaware DOT.

A Delaware DOT spokesman confirmed the stated $12 million, but referred other calls to Amtrak, which he said headed up the project.

In its latest reporting, Amtrak said the Wilmington project supported 176 full-time jobs in the quarter running from October through December.

All told, Delaware got more than $126 million in stimulus funds targeted for Amtrak, or nearly 10 percent of the $1.3 billion Amtrak received from Mr. Obama’s program.

Amtrak said the Wilmington station is the 12th-busiest one in its system, serving nearly 700,000 passengers in 2010.

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