The Fragment Mass Analyzer at the ATLAS facility has been used to unambiguously identify the mass number associated with different decay modes of the nobelium isotopes produced via 204Pb(48Ca,xn)(252-x)No reactions. Isotopically pure (>99.7%) 204Pb targets were used to reduce background from more favored reactions on heavier lead isotopes. Two spontaneous fission half-lives (t_1/2 = 3.7+1.1-0.8 us and 43+22-15 us) were deduced from a total of 158 fission events. Both decays originate from 250No rather than from neighboring isotopes as previously suggested. The longer activity most likely corresponds to a K-isomer in this nucleus. No conclusive evidence for an alpha branch was observed, resulting in upper limits of 2.1% for the shorter lifetime and 3.4% for the longer activity.