Long-lived, heavy particles are predicted in a number of models beyond the standard model of particle physics. We present the first direct search for such particles' decays, occurring up to 100 h after their production and not synchronized with an accelerator bunch crossing. We apply the analysis to the gluino (g), predicted in split supersymmetry, which after hadronization can become charged and lose enough momentum through ionization to come to rest in dense particle detectors. Approximately 410 pb(-1) of pp collisions at square root(s) = 1.96 TeV collected with the D0 detector during Run II of the Fermilab Tevatron collider are analyzed in search of such "stopped gluinos" decaying into a gluon and a neutralino (chi(1)(0)). Limits are placed on the (gluino cross section) x (probability to stop) x [BR(g --> g chi(1)(0))] as a function of the gluino and chi(1)(0) masses, for gluino lifetimes from 30 micros-100 h.