  We prove a version of the Hopf-Rinow-theorem with respect to path metrics on
discrete spaces. The novel aspect is that we do not a priori assume local
finiteness but isolate a local finiteness type condition, called essential
local finiteness, that is indeed necessary. As a side product we identify the
maximal weight, called the geodesic weight, which generates the path metric in
the situation when the space is complete with respect to any of the equivalent
notions of completeness proven in the Hopf-Rinow theorem. As an application, we
characterize the graphs for which the resistance metric is a path metric
induced by the graph structure.
