We report the observation of bottom-charmed mesons B_c in 1.8 TeV ppbar collisions using the CDF detector at the Fermilab Tevatron. The B_c mesons were found through their semileptonic decays, B^+/-_c -> J/psi l^+/- X. A fit to the J/psi l mass distribution yielded 20.4^+6.2_ -5.5 events from B_c mesons. A test of the null hypothesis, i.e. an attempt to fit the data with background alone, was rejected at the level of 4.8 standard deviations. By studying the quality of the fit as a function of the assumed B_c mass, we determined M(B_c) = 6.40 +/- 0.39(stat.) +/- 0.13(syst.) GeV/c^2. From the distribution of trilepton intersection points in the plane transverse to the beam direction we measured the B_c lifetime to be tau(B_c) = 0.46^+0.18_ -0.16(stat.) +/- 0.03(syst.) ps. We also measured the ratio of production cross section times branching fraction for B^+_c -> J/psi l^+ nu relative to that for B^+ -> J/psi K^+ to be: sigma(B_c) . BR(B_c -> J/psi l nu)/sigma(B) . BR(B -> J/psi K) = 0.132^+0.041_ -0.037(stat.) +/- 0.031(syst.)^+0.032/_ -0.020(lifetime)