  We present results for lattice QCD with staggered fermions in the limit of
infinite gauge coupling, obtained from a worm-type Monte Carlo algorithm on a
discrete spatial lattice but with continuous Euclidean time. This is achieved
by sending both the anisotropy parameter $\gamma^2\simeq a/\at$ and the number
of time-slices $N_\tau$ to infinity, keeping the ratio $\gamma^2/N_\tau \simeq
aT$ fixed. In this limit, ambiguities arising from the anisotropy parameter
$\gamma$ are eliminated and discretization errors usually introduced by a
finite temporal lattice extent $\Nt$ are absent. The obvious gain is that no
continuum extrapolation $N_\tau \rightarrow \infty$ has to be carried out.
Moreover, the algorithm is faster and the sign problem disappears completely.
  As a first application, we determine the phase diagram as a function of
temperature and real and imaginary baryon chemical potential. We compare our
computations with those on lattices with discrete Euclidean time.
Discretization errors due to finite $\Nt$ in previous studies turn out to be
large at low temperatures.
