  In 1970, Plummer defined a well-covered graph to be a graph $G$ in which all
maximal independent sets are in fact maximum. Later Hartnell and Rall showed
that if the Cartesian product $G \Box H$ is well-covered, then at least one of
$G$ or $H$ is well-covered. In this paper, we consider the problem of
classifying all well-covered Cartesian products. In particular, we show that if
the Cartesian product of two nontrivial, connected graphs of girth at least $4$
is well-covered, then at least one of the graphs is $K_2$. Moreover, we show
that $K_2 \Box K_2$ and $C_5 \Box K_2$ are the only well-covered Cartesian
products of nontrivial, connected graphs of girth at least $5$.
