The Scot tested positive for the prohibited substance, methamphetamine, a stimulant that could aid an athlete's performance because it improves reaction times.
The 28-year-old denies knowingly taking any banned substances.
But that will be no defence if a second test shows traces of the drug, which is commonly known as 'speed'.
The British Olympic Association issued a statement on Tuesday saying it had been notified by the International Olympic Committee (IOC) that one of its athletes had produced a positive sample.
Baxter, Britain's first ever alpine skiing medallist, then released a statement through the BOA to ptotest his innocence.
"I have never knowingly taken any medicine or substance to improve my performance"
He said: "Late on Friday afternoon, I was telephoned by Simon Clegg, team GB's chef de mission, who told me that he had been advised by the IOC's Inquiry Commission that the sample I had produced following my slalom performance at the Games contained traces of a banned substance.
"Naturally I am devastated by this news.
"I have therefore decided to make public that I am the athlete in question because I have never knowingly taken any medicine or substance to improve my performance and, as such, believe that I am entirely innocent.
"I am now working with lawyers and medical experts to present my case to the IOC's Inquiry Commission (and the IOC's Disciplinary Commission), which I am advised is unlikely to convene until next week, with a view to defending myself successfully against the charges of doping.
"I do not intend to conduct my defence through the media and, as such, I am not making myself available for interview and will not be making any further statement on this matter until the IOC's Inquiry and Disciplinary Commissions have ruled on my case."
The BOA said its primary concern was for the welfare and defence of the athlete and that it would not be making any further comment or involving itself in any speculation until the due process was complete.
The IOC revealed on Tuesday that two athletes failed drugs tests on the final weekend at Salt Lake City.
However, the other athlete has not come forward and his or her identity remains unknown.
It is unusual for alpine skiing to suffer from a drugs scandal, but the cross-country skiing competitions had already been affected by failed drugs tests.
Russia's Larissa Lazutina and Spain's Johann Muehlegg were both stripped of their gold medals after testing positive for Darbepoetin.
If Baxter fails to clear his name he too could suffer the ignominy of being stripped of his medal.