We report the first measurement of the exclusive cross sections e+e− → BB¯ , e+e− → BB¯ ∗, and e+e− → B∗B¯ ∗ in the energy range from 10.63 GeV to 11.02 GeV. The B mesons are fully reconstructed in a large number of hadronic final states and the three channels are identified using a beam-constrained-mass variable. The shapes of the exclusive cross sections show oscillatory behavior with several maxima and minima. The results are obtained using data collected by the Belle experiment at the KEKB asymmetric-energy e+e− collider. [Figure not available: see fulltext.]