We present the results of an extensive exploration of the five-dimensional parameter space of the minimal SU(5) supergravity model, including the constraints of a long enough proton lifetime (${\ensuremath{\tau}}_{p}>1\ifmmode\times\else\texttimes\fi{}{10}^{32}$ yr) and a small enough neutralino cosmological relic density (${\ensuremath{\Omega}}_{\ensuremath{\chi}}{h}_{0}^{2}\ensuremath{\le}1$). We find that the combined effect of these two constraints is quite severe, although still leaving a small region of parameter space with ${m}_{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g},\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{q}}<1$ TeV. The allowed values of the proton lifetime extend up to ${\ensuremath{\tau}}_{p}\ensuremath{\approx}1\ifmmode\times\else\texttimes\fi{}{10}^{33}$ yr and should be fully explored by the SuperKamiokande experiment. The proton lifetime cut also entails the following mass correlations and bounds: ${m}_{h}\ensuremath{\lesssim}110$ GeV, ${m}_{\ensuremath{\chi}}\ensuremath{\approx}\frac{1}{2}{m}_{{\ensuremath{\chi}}_{2}^{0}}\ensuremath{\approx}0.15{m}_{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}$, ${m}_{{\ensuremath{\chi}}_{2}^{0}}\ensuremath{\approx}{m}_{{\ensuremath{\chi}}_{1}^{+}}$, and ${m}_{\ensuremath{\chi}}<85(115)$ GeV, ${m}_{{\ensuremath{\chi}}_{2}^{0},{\ensuremath{\chi}}_{1}^{+}}<165 (225)$ GeV for ${\ensuremath{\alpha}}_{3}=0.113 (0.120)$. Finally, the combined proton decay and cosmology constraints predict that if ${m}_{h}\ensuremath{\gtrsim}75(80)$ GeV then ${m}_{{\ensuremath{\chi}}_{1}^{+}}\ensuremath{\lesssim}90(110)$ GeV for ${\ensuremath{\alpha}}_{3}=0.113(0.120)$. Thus, if this model is correct, at least one of these particles will likely be observed at the CERN ${e}^{+}{e}^{\ensuremath{-}}$ collider LEP II.