Weymouth were relegated from the Blue Square Premier on Friday evening with defeat at home to York sending them down.
Richard Brodie gave the visitors to the Wessex Stadium the lead after just 14 minutes to give the hosts that sinking feeling, but Cliff Akurang got Weymouth back into the game on the stroke of half time.
The home side battled hard for a winner but it was The Minstermen who struck next, as Adam Boyes netted.
Weymouth worked hard for an opening, but were unable to bag even an equaliser as they eventually lost their fight for survival.