The King Center announced the sad news Monday -- noting Dexter succumbed to prostate cancer ... which he seems to have been dealing with privately behind the scenes. The org added his wife, Leah Weber King, was with him during his death ... and she said, "He transitioned peacefully in his sleep at home with me in Malibu."

It's unclear when Dexter was diagnosed -- but he is now the second child of MLK to have passed since their father was assassinated in 1968.

Yolanda King was the first of the children to die -- she passed in '07 as a result of complications from ovarian cancer, among other ailments -- and now ... Dexter has passed on too. The remaining living kids are Martin Luther King III (the eldest) and Bernice King (the youngest out of all of them).

Dexter continued his father's legacy in fighting for civil rights into the 21st century -- but he was notable for confronting James Earl Ray -- the man convicted and imprisoned for actually assassinating Dr. King -- and asking him point blank if he did it ... only to receive a "no."

At the time, DSK said he actually believed James ... and floated conspiracy theories about who really orchestrated his father's murder, suggesting the federal government was involved.

Dexter also portrayed his dad in a 2002 TV movie, 'The Rosa Parks Story' -- it goes without saying, his resemblance to Dr. King was something that was always talked about.

He was only 62.