The Clark County Office of the Coroner/Medical Examiner tells us that the cause of death has been ruled an accident for the "Pawn Stars" personality's son, who died in Las Vegas last month.

TMZ was the first to report Adam's passing ... after a rep for RH shared that Adam had died from a fatal OD. The Harrison family told TMZ shortly after his passing ... "Our family is extremely saddened by the death of Adam. We ask for privacy as we grieve his loss." At the time -- the Las Vegas Metro PD told us they were investigating the case.

As we reported ... Adam was struggling in the days leading up to his death -- with former roommates telling cops that AH was displaying erratic and strange behavior before allegedly barricading himself inside a guest house where he was staying. He reportedly had just been released from jail after serving 3 months behind bars -- although we don't know the cause for his incarceration.

Adam -- one of Rick's three boys -- kept mostly out of the limelight ... having stepped away from helping out at the World Famous Gold & Silver Pawn Shop. Adam is Rick's kid from his first marriage to ex-wife Kim, with whom he also shares son Corey.

The "Pawn Stars" leading man later welcomed son Jake with his second wife, Tracy. Corey and Jake are regulars on his reality show -- but Adam did not appear on camera.

On losing his son, Rick shared on Instagram, "You will always be in my heart! I love you Adam."

Adam was 39.