The German Olympic Sports Confederation (DOSB) said it had been informed by the International Olympic Committee (IOC) that the "A" sample "of a member of the German Olympic team produced a result that diverged from the norm".
The "B" sample is to be tested on Friday while the IOC's disciplinary commission will meet to discuss the issue, the DOSB said in a statement on its website.
There were no details on the name or sport of the athlete, or whether the test had been carried out at the Games or before.
The IOC confirmed the content of the German statement but said no further information would be given until the investigation is over.
The last Winter Olympics in Vancouver 2010 produced only one positive doping test in the course of the Games.
Slovak ice hockey player Lubomir Visnovsky tested positive for a banned stimulant he consumed in a cold medication. He was issued with a reprimand but not banned.
Later in 2010, Polish cross country skier Kornelia Marek tested positive on a sample taken during the Games. She was given a two-year ban.
The IOC, which oversees drug testing at the Olympic Games, is carrying out almost 2,500 drug tests at Sochi 2014 with an extra emphasis on out-of-competition tests.
The Russian team was hit by a major scandal on the eve of the Sochi Games when one of its top female biathletes, Irina Starykh, withdrew from the Olympics when an "A" sample tested positive for an unspecified banned substance.
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