This Letter presents the first strong evidence for the resolution of the excited B mesons B(1) and B(2)* as two separate states in fully reconstructed decays to B(+)(*)pi(-). The mass of B(1) is measured to be 5720.6+/-2.4+/-1.4 MeV/c(2) and the mass difference DeltaM between B(2)* and B(1) is 26.2+/-3.1+/-0.9 MeV/c;{2}, giving the mass of the B(2)* as 5746.8+/-2.4+/-1.7 MeV/c(2). The production rate for B(1) and B(2)* mesons is determined to be a fraction (13.9+/-1.9+/-3.2)% of the production rate of the B+ meson.