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 1 x 10(32) cm(-2) s(-1), an integrated luminosity of 10 pb(-1), 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 m((g) over tilde) - m((chi) over tilde1)(0) >100 GeV/c(2), and assuming BR((g) over tilde -> g<(chi over bar>(0)(1)) = 100%, m((g) over tilde) < 370 GeV/c(2) are excluded for lifetimes from 10 mu s to 1000 s.