The decision by the International Olympic Committee's executive committee was confirmed at a news conference on Thursday.
Baxter - the first Briton ever to win an Olympic skiing medal - had failed a drugs test in Salt Lake City.
In a statement the IOC said Baxter had been disqualified and ordered to return his slalom medal within 10 days.
"I have to speak to my lawyers and see - I think we have a really good case"
Alain Baxter on the prospects of an appeal
Craig Reedie, chairman of the British Olympic Association, said he was "very disappointed at the decision reached by the IOC".
"The BOA is convinced that in no way can Alain be described as a drugs cheat," Reedie said.
"We believe the offence to be modest and the punishment very severe."
And Baxter, who had not spoken publicly since the episode began, announced he was considering launching an appeal against the disqualification.
"I have to speak to my lawyers and see," he said. "I think we have a really good case but I'm not sure what is happening."
The bronze medal will now go to Benjamin Raich of Austria who finished fourth. Raich also won a bronze in the men's combined event at the Games.
Baxter revealed how he had been in the midst of celebrating in his home village of Aviemore in Scotland when he found out about his positive test for the banned substance methamphetamine.
"The homecoming was one of the most amazing experiences of my life," he said. "Such a small village with such support behind me.
"Two days later I took a call saying I had tested positive. I had no idea what this positive sample could be, and three weeks later here I am now."
Baxter has received an automatic ban from competition skiing which lasts until June.
The International Skiing Federation have to decide now whether to extend that ban, as they could prolong it to as much as two years.
"He's currently banned until June by the International Skiing Federation," said a spokesperson for the British Ski and Snowboard Federation (BSSF).
The BSSF spokesperson added: "They have to decide whether to extend his ban, but we're hoping that it will be no more than three months when you consider how unintentional this was."
Baxter admitted: "I'm just hoping it won't be two years. I've no idea what is happening."
An emotional Baxter reiterated the point he made to the IOC last week - that the positive drugs test came from an over-the-counter nasal inhaler which he bought in the US.
But the IOC's strict liability rule meant he was responsible for the substance in his body.
It was widely expected the 28-year-old would lose the bronze after he failed a drugs test.
Baxter, from Aviemore in Scotland, tested positive for traces of methamphetamine after his amazing performance in the men's slalom at Salt Lake City on 23 February.
He spent two days at the IOC headquarters in Lausanne last week arguing his case with a team of lawyers.
He was also backed there by the British Olympic Association and the British Ski and Snowboard Federation, but to no avail.