  We derive an annealed large deviation principle for the normalised local
times of a continuous-time random walk among random conductances in a finite
domain in $\Z^d$ in the spirit of Donsker-Varadhan \cite{DV75}. We work in the
interesting case that the conductances may assume arbitrarily small values.
Thus, the underlying picture of the principle is a joint strategy of small
values of the conductances and large holding times of the walk. The speed and
the rate function of our principle are explicit in terms of the lower tails of
the conductance distribution. As an application, we identify the logarithmic
asymptotics of the lower tails of the principal eigenvalue of the randomly
perturbed negative Laplace operator in the domain.
