We report on the first observation of the bottom-charm mesons B[sub c] through the decay mode B[sub c][sup[+-]][yields] J/[psi]l[sup[+-]][nu] in 1.8 TeV p[anti p] collisions using the CDF detector at the Fermilab Tevatron. A fit of background and signal contributions to the observed J/[psi]l mass distribution yielded 20.4[sup+6.2][sub -5.5] events from B[sub c] mesons. A fit to the same distribution with background alone was rejected at the level of 4.8 standard deviations. We measured the B[sub c][sup+] mass to be 6.40[+-]0.39[+-]0.13 GeV/c[sup 2] and the B[sub c][sup+] lifetime to be 0.46[sup 0.18][sub -0.16](stat.)[+-]0.03(syst.) ps. The measured production cross section times branching ratio for B[sub c][sup+][yields] J/[psi]l[sup+][nu] relative to that for B[sup+][yields] J/[psi]K[sup+] is 0.132[sup 0.041][sub -0.037](stat.)[+-]0.031(syst.)[sup+0.032][sub -0.020](lifetime).