The Chelsea midfielder, 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 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 remains unclear whether any other members of the squad must also isolate as close contacts.