  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 hours 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 p-pbar collisions at 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 (~X_0), reconstructed as a jet and missing energy. No
excess is observed above background, and limits are placed on the (gluino cross
section) x (probability to stop) x BR(~g -> g ~X_0) as a function of the gluino
and ~X_0 masses, for gluino lifetimes from 30 $\mu$s -- 100 hours.
