  We propose a new experiment to measure the running of the fine-structure
constant in the space-like region by scattering high-energy muons on atomic
electrons of a low-Z target through the process $\mu e \to \mu e$. The
differential cross section of this process, measured as a function of the
squared momentum transfer $t=q^2<0$, provides direct sensitivity to the
leading-order hadronic contribution to the muon anomaly $a^{\rm{HLO}}_{\mu}$.
By using a muon beam of 150 GeV, with an average rate of $\sim1.3\times 10^7$
muon/s, currently available at the CERN North Area, a statistical uncertainty
of $\sim 0.3\%$ can be achieved on $a^{\rm{HLO}}_{\mu}$ after two years of data
taking. This direct measurement of $a^{\rm{HLO}}_{\mu}$ will provide an
independent determination, competitive with the time-like dispersive approach,
and consolidate the theoretical prediction for the muon $g$-2 in the Standard
Model. It will allow therefore a firmer interpretation of the measurements of
the future muon $g$-2 experiments at Fermilab and J-PARC.
