NEW YORK – Police may be on the brink of finally finding the remains of Judge Joseph Crater – and possibly solving the most enduring mystery in New York City history, investigators say.

A letter probably penned more than half a century ago has pinpointed a spot under the Coney Island boardwalk where Crater – last seen leaving a Times Square watering hole 75 years ago – is buried, they said.

The letter was written by an elderly woman who died of natural causes in June, they said.

Upon her death, family members opened a safe-deposit box, where they found a letter with “Do Not Open Until My Death” instructions on the envelope.

In it, the woman claimed that her father told her on his deathbed where Crater’s remains were and that cab driver Frank Burn was his killer.

The letter also mentions Burn’s brother and a city cop named Charles.

The names of the woman and her father are being kept secret.

Detectives said everything the woman wrote so far has been corroborated.

For generations, Joseph Crater – a Tammany Hall stalwart, justice of the state Supreme Court and dapper man about town – has been what some newspapers have called “the most missingest man in America.”

His baffling disappearance spawned thousands of jokes and from time to time rumors circulated of him being spotted on a South Sea isle or in some other remote locale.

Appointed to the trial court bench by then-New York Gov. Franklin Roosevelt, Crater, 41, had been a judge just four months when he vanished.

On Aug. 6, 1930, he cashed two checks amounting to more than $5,000 and took along an additional $20,000 in campaign money – equal to about $250,000 today.

He hastily packed up files in his office and, as he left his chambers, told his assistant, “Don’t forget to turn out the lights, Johnson.”

That evening, the tall, 200-pound judge, with slicked-back, graying hair parted down the middle, dined with his showgirl mistress and another friend.

After dinner, “Good Time Joe” – his moniker because he loved to dance – hailed a cab, waved his straw Panama hat out the window and vanished without a trace.

A grand jury called 95 witnesses and amassed nearly 1,000 pages of testimony, but never learned what happened to him.

Some have speculated that the dapper jurist was rubbed out by – or at the request of – his girlfriend’s other boyfriend, a mobster.

Two months after Crater vanished, the girlfriend also disappeared.

Others have long suspected that notorious mobster Frank Costello offed Crater as payment to Tammany Hall pols who were protecting his rackets.

Still others believed Crater was killed because he was about to expose corruption in Tammany Hall.

Investigators who have examined erosion records think they’ve zeroed in on where Crater’s remains may be, but they haven’t said when digging will start.

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