For the first time a direct determination of the atomic mass difference of Re187 and Os187 has been performed with the Penning-trap mass spectrometer SHIPTRAP applying the novel phase-imaging ion-cyclotron-resonance technique. The obtained value of 2492(30stat)(15sys) eV is in excellent agreement with the Q values determined indirectly with microcalorimetry and thus resolves a long-standing discrepancy with older proportional counter measurements. This is essential for the determination of the neutrino mass from the β- decay of Re187 as planned in future microcalorimetric measurements. In addition, an accurate mass difference of Re187 and Os187 is also important for the assessment of Re187 for cosmochronology.