We study the frame property of the Gabor system of time-frequency shifts

$$\mathfrak{G}(g; \alpha, \beta) := \{ e^{2\pi i \beta\beta mx} g(x-n\alpha) \}_{m,n \in \mathbb{Z}}$$

generated by rational windows $g$. In particular we prove that for Herhlotz functions $g$ the system $\mathfrak{G}(g; \alpha, \beta)$ constitutes a frame in $L^2(\mathbb{R})$ for all lattices $\alpha \mathbb{Z} \times \beta \mathbb{Z}$ of density at least 1, $\alpha\beta \leq 1$. In addition, we prove that for general rational windows $g$ the Gabor system $\mathfrak{G}(g; \alpha, \beta)$ is a frame for all irrational densities $\alpha\beta < 1$.