Weems, the 255th overall selection out of Baylor University, hit .267 with seven home runs and 30 RBIs this spring for the Bears. He was a unanimous first-team Big 12 pick during the 2007 season and an honorable mention selection in 2008.
Weems' 24 career home runs tied the school record for shortstops.
Weems will likely report to short-season Eugene of the Northwest League to begin his professional career.
The Padres still haven't signed their top pick, former Wake Forest first baseman Allan Dykstra.
The Padres and Dykstra had originally agreed on a $1.4 million signing bonus, although the team backed off after looking into a hip condition Dykstra had that required surgery in high school. Dykstra is still expected to sign at some point.
With the signing of Weems, the Padres have now signed nine of their first 15 Draft picks, including two first-round compensatory picks, outfielder Jaff Decker and infielder Logan Forsythe.
Corey Brock is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.