  We study the $L^2$-critical damped NLS with a Stark potential. We prove that
the threshold for global existence and finite time blowup of this equation is
given by $\|Q\|_2$, where $Q$ is the unique positive radial solution of $\Delta
Q + |Q|^{4/d} Q = Q$ in $H^1(\mathbb{R}^d)$. Moreover, in any small
neighborhood of $Q$, there exists an initial data $u_0$ above the ground state
such that the solution flow admits the log-log blowup speed. This verifies the
structural stability for the ``$\log$-$\log$ law'' associated to the NLS
mechanism under the perturbation by a damping term and a Stark potential. The
proof of our main theorem is based on the Avron-Herbst formula and the
analogous result for the unperturbed damped NLS.
