If is a finite-dimensional associative k-algebra with unit, then its rank R(), i.e. its bilinear complexity, is never less than . is said to be of minimal rank if . In this paper we determine for infinite perfect fields k all commutative k-algebras of minimal rank. Roughly speaking, these algebras are built up from simply generated structures which annihilate each other. Furthermore, we indicate how this result can be used to obtain new lower bounds for the rank of specific commutative algebras.