  We investigate the possibility of observing the exotic decay mode of the top
quark into the lightest stop ($\tilde t_1$) and neutralino ($\tilde\chi^0_1$)
in the minimal supersymmetric standard model with R-parity at the upgraded
Tevatron. First we determine the allowed range for the branching fraction
$B(t\to \tilde t_1 \tilde\chi^0_1)$ in the region of parameter space allowed by
the $R_b$ data and the CDF $ee\gamma\gamma+{\large \not} E_T$ event, and then
consider all possible backgrounds and investigate the possibility of observing
this final state at the Tevatron. We find that this final state is unobservable
at Run 1. However, Run 2 can provide significant information on this new decay
mode of the top quark: either discover it, or establish a strong constraint on
the masses of $\tilde t_1$ and $\tilde\chi^0_1$ given approximately by
$M_{\tilde \chi^0_1} > M_{\tilde t_1} - 6$ GeV.
