Chelsea midfielder Gilmour, who was outstanding in his first international start in the draw with England on Friday night, must now isolate for 10 days.
It means he is out of the final game of Group D at Hampden Park, a match Scotland need a positive result in to qualify for the knockout phase.
“The Scottish FA can confirm that a member of the Scotland National Team playing squad, Billy Gilmour, has tested positive for Covid-19,” a statement from the Scottish Football Association on Monday read.
“Having liaised with Public Health England since the positive test was recorded, Billy will now self-isolate for 10 days and therefore miss tomorrow’s Uefa Euro 2020 Group D match against Croatia at Hampden.”
It is understood that no other members of the squad or staff must isolate as close contacts.
The FA have confirmed that all 26 England players and the wider support team returned negative results after the latest round of testing on Sunday and are continuing to follow appropriate Covid-19 protocols.
Gilmour impressed at Wembley and was in line to keep his place against Croatia on Tuesday night.
Steve Clarke’s side, currently fourth in Group D, must win to stand any chance of qualifying for the last 16.