Experimental searches for neutrinoless double-β decay offer one of the best opportunities to look for physics beyond the standard model. Detecting this decay would confirm the Majorana nature of the neutrino, and a measurement of its half-life can be used to determine the absolute neutrino mass scale. Important to both tasks is an accurate knowledge of the Q value of the double-β decay. The LEBIT Penning trap mass spectrometer was used for the first direct experimental determination of the Zr96 double-β decay Q value: Qββ=3355.85(15) keV. This value is nearly 7 keV larger than the 2012 Atomic Mass Evaluation [M. Wang, Chin. Phys. C 36, 1603 (2012)1674-113710.1088/1674-1137/36/12/003] value and one order of magnitude more precise. The 3-σ shift is primarily due to a more accurate measurement of the Zr96 atomic mass: m(96Zr)=95.90827735(17) u. Using the new Q value, the 2νββ-decay matrix element, |M2ν|, is calculated. Improved determinations of the atomic masses of all other zirconium (Zr90-92,94,96) and molybdenum (92,94-98,100Mo) isotopes using both 12C8 and Rb87 as references are also reported.