Using approximately 1.3 fb-1 of data collected by the D0 detector between 2002 and 2006, we measure the lifetime of the B_c meson in the B_c -> J/psi mu nu X final state. A simultaneous unbinned likelihood fit to the J/\psi+mu invariant mass and lifetime distributions yields a signal of 881 +/- 80 (stat) candidates and a lifetime measurement of \tau(B_c) = 0.448 +0.038 -0.036 (stat) +/- 0.032 (syst) ps.