  We present a scheme to estimate Gaussian states of one-dimensional continuous
variable systems, based on weak (unsharp) quantum measurements. The estimation
of a Gaussian state requires us to find position ($q$), momentum ($p$) and
their second order moments. We measure $q$ weakly and follow it up with a
projective measurement of $p$ on half of the ensemble, and on the other half we
measure $p$ weakly followed by a projective measurement of $q$. In each case we
use the state twice before discarding it. We compare our results with
projective measurements and demonstrate that under certain conditions such weak
measurement-based estimation schemes, where recycling of the states is
possible, can outperform projective measurement-based state estimation schemes.
