  In the present paper the cyclic homology functor from the category of
$A_\infty$-algebras over any commutative unital ring $K$ to the category of
graded $K$-modules is constructed. Further, it is showed that this functor
sends homotopy equivalences of $A_\infty$-algebras into isomorphisms of graded
modules. As a corollary, it is obtained that the cyclic homology of an
$A_\infty$-algebra over any field is isomorphic to the cyclic homology of the
$A_\infty$-algebra of homologies for the source $A_\infty$-algebra.
