  In this paper we consider a heavy gluino to be the lightest supersymmetric
particle [LSP]. We investigate the limits on the mass of a heavy gluino LSP,
using the searches for excess events in the jets plus missing momentum channel
in Run I. The neutral and charged R-hadrons, containing a heavy gluino LSP,
have distinct signatures at the Fermilab Tevatron. The range of excluded gluino
masses depends on whether the R-hadron is charged or neutral and the amount of
energy deposited in the hadronic calorimeter. The latter depends on the energy
loss per collision in the calorimeter and the number of collisions; where both
quantities require a model for R-hadron- Nucleon scattering. We show how the
excluded range of gluino mass depends on these parameters. We find that gluinos
with mass in the range between $\sim 35$ GeV and $\sim 115$ GeV are excluded by
CDF Run I data. Combined with previous results of Baer et al., which use LEP
data to exclude the range 3 - 22$\sim$25 GeV, our result demonstrates that an
allowed window for a heavy gluino with mass between 25 and 35 GeV is quite
robust. Finally we discuss the relevant differences of our analysis of Tevatron
data to that of Baer et al.
