  We calculate the quantum Cram\'er--Rao bound for the sensitivity with which
one or several parameters, encoded in a general single-mode Gaussian state, can
be estimated. This includes in particular the interesting case of mixed
Gaussian states. We apply the formula to the problems of estimating phase,
purity, loss, amplitude, and squeezing. In the case of the simultaneous
measurement of several parameters, we provide the full quantum Fisher
information matrix. Our results unify previously known partial results, and
constitute a complete solution to the problem of knowing the best possible
sensitivity of measurements based on a single-mode Gaussian state.
