We report the discovery of a morphology-density relation for clusters of galaxies analogous to the dependence of galaxy types on local galaxy density. The relationship is such that flattened and linear clusters (Rood-Sastry types F and L) occur predominantly in rich supercluster environments, while rounder or more amorphous clusters (Rood-Sastry types C and I) are much more prevalent in low-density regions. A clear trend in the relative fractions of C + I and L + F clusters is seen as one progresses to denser and denser regions. The occurrence of cD and B type clusters, on the other hand, is independent of local cluster density. Bautz-Morgan classifications also show no correlation with cluster environment. If one adopts the usual interpretation of the Rood-Sastry classification scheme as corresponding to different dynamical states of clusters, with I and F clusters being less dynamically evolved than C and L, which in turn are less dynamically evolved than cD and B clusters, then the existence of this morphology-density relation suggests that it is the shape of the cluster potential, rather than the time scale for cluster collapse, that is most environment-dependent. Furthermore, the fact that in all environments ∼ 30%-50% of clusters have unrelaxed morphologies also indicates that cluster formation must be a recent occurrence. The lack of any morphology-density relation for Rood-Sastry cD and B clusters or for Bautz-Morgan types suggests that these classifications represent more a measure of the evolution of the luminosity function of galaxies within a cluster rather than a true indicator of the cluster's overall dynamical state.