  The MUonE experiment has been proposed to measure the differential cross
section of $\mu e$ elastic scattering, by colliding the 160 GeV muons of the
CERN M2 beam with atomic electrons of thin target plates. From a very precise
measurement of the shape one can achieve a competitive determination of the
leading hadronic contribution to the muon magnetic moment, independent from the
other existing ones. In preparation for the Test Run with a reduced setup the
detector geometry has been optimised. Expected yields for a first physics run
with limited statistics are discussed, together with prospects for the
assessment of the main systematic uncertainties.
