  Let $\h_n$ be the Cartan subalgebra of the Witt algebras
$\W_n^+=\text{Der}\C[t_1, t_2, ..., t_n]$ and $\W_n=\text{Der}\C[t_1^{\pm
1},t_2^{\pm 1},\cdots,t_n^{\pm1}]$ where $1\le n\le \infty$. In this paper, we
classify the modules over $\W_n^+$ and over $\W_n$ which are free
$U(\h_n)$-modules of rank $1$. These are the $\W_n^+$-modules
$\Omega(\Lambda_{n},a, S) $ for some $\Lambda_n=(\lambda_1,\cdots,\lambda_n)
\in (\C^*)^n, a\in \C$, and $S\subset \{1,2,..., n\}$; and $\W_n$-modules
$\Omega(\Lambda_n,a)$ for some $\Lambda_n\in (\C^*)^n$ and some $a\in \C.$
