  We consider a random walk on the discrete cylinder
$({\mathbb{Z}}/N{\mathbb{Z}})^d\times{\mathbb{Z}}$, $d\geq3$ with drift
$N^{-d\alpha}$ in the $\mathbb{Z}$-direction and investigate the large
$N$-behavior of the disconnection time $T^{\mathrm{disc}}_N$, defined as the
first time when the trajectory of the random walk disconnects the cylinder into
two infinite components. We prove that, as long as the drift exponent $\alpha$
is strictly greater than 1, the asymptotic behavior of $T^{\mathrm{disc}}_N$
remains $N^{2d+o(1)}$, as in the unbiased case considered by Dembo and
Sznitman, whereas for $\alpha<1$, the asymptotic behavior of
$T^{\mathrm{disc}}_N$ becomes exponential in $N$.
