We show that in addition to the photinolike and Higgsino-like states usually considered as candidates for the lightest supersymmetric particle (LSP), there is yet one more state in the neutralino sector which could be the LSP in minimal supergravity models. In this paper, we study how the phenomenology of supersymmetry is altered depending on the nature of the LSP. Within the framework of minimal supergravity models, we demonstrate that if the mass of the LSP is small compared to that of the W boson, the lightest chargino and the next to lightest neutralino states are always lighter than ${M}_{W}$ and ${M}_{Z}$ independent of the soft-supersymmetry-breaking gaugino masses. We show that the W- and Z-boson decay widths into these particles depends quite significantly on the nature of the LSP. We further show that the bound on the scalar-electron mass of the ASP detector at the SLAC storage ring PEP is considerably weakened if the LSP is the new state discussed above, even if this state is dominantly a gaugino. Finally, we briefly study the phenomenology of the LSP in the context of superstring-inspired E(6) models.