The authors prove the conjecture that every finite-dimensional Hopf algebra H over a field k is a free (left or right) module over any Hopf subalgebra B. This was first proved by the second author in the case when \(B=kG\), G the group of group-like elements, is semi-simple [J. Algebra 118, 102-108 (1988; Zbl 0649.16007)], and then both authors proved it for \(B=kG\) in general [J. Pure Appl. Algebra 56, 51-57 (1989; Zbl 0659.16006)]. Actually, a somewhat more general result is proved for finite-dimensional H, namely that every left (H,B)-Hopf module is free as a left B-module. The result is known to be false for infinite-dimensional H, but is true if H is commutative and B is finite-dimensional, as shown by \textit{D. E. Radford} [J. Pure Appl. Algebra 11, 15-28 (1977; Zbl 0367.16004)].