For $\textbf{r}=(r_1,\ldots,r_k)$, an $\textbf{r}$-factorization of the complete $λ$-fold $h$-uniform $n$-vertex hypergraph $λK_n^h$ is a partition of the edges of $λK_n^h$ into $F_1,\ldots, F_k$ such that $F_j$ is $r_j$-regular and spanning for $1\leq j\leq k$. This paper shows that for $n>\frac{m-1}{1-2^{\frac{1}{1-h}}}+h-1$, a partial $\textbf{r}$-factorization of $λK_m^h$ can be extended to an $\textbf{r}$-factorization of $λK_n^h$ if and only if the obvious necessary conditions are satisfied.