  For any triple $(\mu,\lambda,\alpha)$ of complex numbers and an $\mathfrak
a$-module ${V}$, a class of non-weight modules
$\mathcal{M}\big(V,\mu,\Omega(\lambda,\alpha)\big)$ over the Virasoro algebra
$\mathcal L$ is constructed in this paper. We prove if $V$ is a nontrivial
simple $\mathfrak a$-module satisfying: for any $v\in V$ there exists
$r\in\Z_+$ such that $L_{r+i}v=0$ for all $i\geq1$, then
$\mathcal{M}\big(V,\mu,\Omega(\lambda,\alpha)\big)$ is simple if and only if
$\mu\neq1, \lambda\neq0,\alpha\neq0,$. We also give the necessary and
sufficient conditions for two such simple $\mathcal L$-modules being
isomorphic. Finally, we prove that these simple $\mathcal L$-modules
$\mathcal{M}\big(V,\mu,\Omega(\lambda,\alpha)\big)$ are new by showing they are
not isomorphic to any other known simple non-weight module provided that $V$ is
not a highest weight $\mathfrak a$-module with highest weight nonzero.
