Answer:
In the movie they don't mention the cause of death, but in the book it says that he collapsed and died of pneumonia: "The coroner announced...that Williams appeared to have died of natural causes, most likely a heart attack. After an autopsy, the coroner was more specific: Williams had died of pneumonia. This started another rumor -- that he might have died of AIDS. But Williams had shown no signs of being ill...
"...Minerva, of course, had her own idea about what had happened. 'It was the boy that done it,' she said. A little-noticed detail of Williams's death lent an eerie ring of truth to her pronouncement...He had been found lying on the carpet behind the desk in the very spot where he would have fallen eight years earlier, if Danny Hansford had actually fired a gun and the shots had found their mark." (Random House hardcover edition, pp 384 - 385).