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 ($\stackrel{\texttildelow{}}{g}$), 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 $410\text{ }\text{ }{\mathrm{pb}}^{\ensuremath{-}1}$ of $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$ 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 (${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{0}$). Limits are placed on the $(\mathrm{\text{gluino cross section}})\ifmmode\times\else\texttimes\fi{}(\mathrm{\text{probability to stop}})\ifmmode\times\else\texttimes\fi{}[\mathrm{BR}(\stackrel{\texttildelow{}}{g}\ensuremath{\rightarrow}g{\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{0})]$ as a function of the gluino and ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{0}$ masses, for gluino lifetimes from $30\text{ }\text{ }\ensuremath{\mu}\mathrm{s}--100\text{ }\text{ }\mathrm{h}$.