Using 535 Â 10 6 B-meson pairs collected by the Belle detector at the KEKB e þ e À collider, we measure branching fractions of ð7:16 AE 0:10ðstatÞ AE 0:60ðsystÞ Â 10 À4 for B þ ! J=c K þ þ À and ð4:31 AE 0:20ðstatÞ AE 0:50ðsystÞÞ Â 10 À4 for B þ ! c 0 K þ þ À . We perform amplitude analyses to determine the resonant structure of theis a prominent component of both decay modes. There is significant interference among the different intermediate states, which leads, in particular, to a striking distortion of the line shape due to the !. Based on the results of the fit to the B þ ! J=c K þ þ À data, the relative decay fractions of the K 1 ð1270Þ to K , K!, and K Ã ð892Þ are consistent with previous measurements, but the decay fraction to K Ã 0 ð1430Þ is significantly smaller. Finally, by floating the mass and width of the K 1 ð1270Þ in an additional fit of the B þ ! J=c K þ þ À data, we measure a mass of ð1248:1 AE 3:3ðstatÞ AE 1:4ðsystÞÞ MeV=c 2 and a width of ð119:5 AE 5:2ðstatÞ AE 6:7ðsystÞÞ MeV=c 2 for the K 1 ð1270Þ.