A graph $G$ with a list of colors $L(v)$ and weight $\omega(v)$ for each vertex $v$ is $(L,\omega)$-choosable if one can choose a subset of $\omega(v)$ colors from $L(v)$ for each vertex $v$, such that adjacent vertices receive disjoint color sets. In this paper, we give a necessary and sufficient condition for a weighted path to be $(L,\omega)$-choosable for some lists $L$. Furthermore, we solve the problem of the free-choosability of a cycle.