  We introduce and study the weighted $r$-path ideal of a weighted graph
$G_\omega$, which is a common generalization of Conca and De Negri's $r$-path
ideal for unweighted graphs and Paulsen and Sather-Wagstaff's edge ideal of the
weighted graph. Over a field, we explicitly describe primary decompositions of
these ideals, and we characterize Cohen-Macaulayness of these ideals for trees
(with arbitrary $r$) and complete graphs (for $r=2$).
