A new experiment is proposed to measure the running of the electromagnetic coupling constant in the space-like region by scattering high-energy muons on atomic electrons of a low-Z target. The differential cross section of the elastic process μe → μe provides direct sensitivity to the leading-order hadronic contribution to the muon anomaly aμHLO. It is argued that by using the 150-GeV muon beam available at the CERN North Area, with an average rate of ~ 1.3 × 107 muon/s, a statistical uncertainty of ~ 0.3% can be achieved on aμHLO after two years of data taking. The direct measurement of aμHLO via μe scattering will provide an independent determination 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 JPARC.