  For any two complex numbers $a$ and $b$, $\mathcal{V} ir(a,b)$ is a central
extension of $\mathcal{W}(a,b)$ which is universal in the case $(a,b)\neq
(0,1)$, where $\mathcal{W}(a,b)$ is the Lie algebra with basis $\{L_n,W_n\mid
n\in\Z\}$ and relations $[L_m,L_n]=(n-m)L_{m+n}$, $[L_m,W_n]=(a+n+bm)W_{m+n}$,
$[W_m,W_n]=0$. In this paper, we construct and classify a class of non-weight
modules over the algebra $\mathcal{V} ir(a,b)$ which are free $U(\mathbb{C}
L_0\oplus\mathbb{C} W_0)$-modules of rank $1$. It is proved that such modules
can only exist for $a=0$.
