Volquez, 26, was seen as a rising star in 2008 when he posted a 17-6 record with a 3.21 earned run average. He struck out 206 batters in 196 innings and was selected to the National League All-Star team.
But he made just nine starts in 2009, going 4-2 with a 4.35 ERA, because a series of injuries placed him on the disabled list several times.
His season ended June 2 and he later underwent surgery on his right elbow to repair a torn ligament. He wasn't expected to return to the team until late this season at the earliest.
The suspension officially begins Wednesday.
Volquez issued a statement saying the positive test was an isolated incident and apparently came because of medication he was prescribed in his native Dominican Republic during the off-season for a condition he didn't name.
"Unfortunately, I now know that the medication the physician in the Dominican gave me is one that is often used to treat my condition but is also a banned substance under Major League Baseball's drug policy," Volquez said in the statement. "As a result, I tested positive when I reported to spring training."
Volquez said he won't challenge the suspension.