On 7 March 1942, he was happily kibitzing a skittles game at the Manhattan Chess Club in New York when he collapsed from a stroke. He was taken to Mount Sinai hospital, where he died the next morning. Remarkably, the Cuban's great rival, German-born Emanuel Lasker, had died in that very hospital only a year earlier.

His bitter rival Alekhine wrote on Capablanca's death, "With his death, we have lost a very great chess genius whose like we shall never see again."

Chessville.com:

Another version of history, from Jude Acers's final interview with Grandmaster A. Denker:

THE DAY CAPABLANCA DIED

...................Jude, I was there.  I was at the chess club the day the great Cuban passed.  It was to my left.  There was a giant commotion.  I heard "help...help me with my coat" ..Jose was surrounded by a crew of Wall Street admirers, as always...with flowers in their buttonholes..  THE RIGHT PEOPLE A CHESS GRANDMASTER SHOULD KNOW, ABSOLUTELY MUST KNOW ...he was not disdaining toward the little fellow, the businessman or chess amateur who happened to be quite wealthy to boot...  Over their heads I had seen him touch the pieces moments ago...seized now by a fatal coronary attack, lapsed into a coma and lowered onto a table...then gently to the floor for the emergency people just arriving....I had just heard him say his eternal admonition 'ALWAYS PLAY THE NATURAL MOVE' as he liked at the critical moment of a game...in both Spanish and English rapid fire with a "ha-ha- ha" added in staccato fashion.  Just one more thing.  His hand would do a quick wave movement, maybe with a finger hastily extended for emphasis with "ALWAYS PLAY THE NATURAL MOVE!"...Jude as you get older your mind plays tricks on you, sometimes delightfully so...I flash upon the last strange memory of Capablanca's still hand as he died, remembering him waving it with a smile in life. ...Do you know that the widow  returned for many years, exact calendar date and time and place where he died?  To the minute, sitting quietly for a few hours dressed in black.

A page on Capablanca's widow:

On October 20, 1938 Olga Chagodaeva and Jose Raoul Capablanca were married in New York.

On March 7, 1942 Capablanca suffered a stroke at the Manhattan Chess Club in New York while analyzing a chess game. He was rushed to Mount Sinai Hospital, where he died on March 8, 1942, at the age of 53. Olga relates the story of gazing into the night sky, her eyes focused on a bright star that suddenly disappeared at the moment of Capablanca’s death. General Batista, President of Cuba, took charge of the funeral arrangements, and Capablanca was buried with full honors in Havana.