  For any reductive Lie algebra $\mathfrak{g}$ and commutative, associative,
unital algebra $S$, we give a complete classification of the simple weight
modules of $\mathfrak{g}\otimes S $ with finite weight multiplicities. In
particular, any such module is parabolically induced from a simple admissible
module for a Levi subalgebra. Conversely, all modules obtained in this way have
finite weight multiplicities. These modules are isomorphic to tensor products
of evaluation modules at distinct maximal ideals of $S$. Our results also
classify simple Harish-Chandra modules up to isomorphism for all central
extensions of current algebras.
