The results of the first search for long-lived gluinos produced in 7 TeV pp collisions at the CERN Large Hadron Collider are presented. The search looks for evidence of long-lived particles that stop in the CMS detector and decay in the quiescent periods between beam crossings. In a dataset with a peak instantaneous luminosity of 10^{32} cm^{-2} s^{-1}, an integrated luminosity of 10 inverse picobarns, and a search interval corresponding to 62 hours of LHC operation, no significant excess above background was observed. Limits at the 95% confidence level on gluino pair production over 13 orders of magnitude of gluino lifetime are set. For a mass difference between the gluino and the neutralino greater than 100 GeV/c^2, and assuming a branchi...