The autographs of Frank Sinatra are among the most collectible of the twentieth century. Early in his career, it wasn't difficult to get Sinatra to sign an autograph. As he became more popular, it became much harder. There are a lot of Sinatra autographs from the 1960s up on the market, but most of them are not authentic. Sinatra employed secretaries to sign his fan mail. In public it was next to impossible to get close enough to him to get an autograph. Also, as his autograph grew in value, signing would be like giving away money.
If you don't plan on selling, keep a copy of the story with the record. If you do want to sell, search the net for auction houses. They will be happy to sell an authentic Frank Sinatra autograph.
The record is worth more than $1000. It could be worth much more than that, depending on the quality of the autograph.