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We have searched for two-body charmless hadronic decays of B mesons. Final states include ππ, Kπ, and KK with both charged and neutral kaons and pions; πρ, Kρ, and K * π; and Kφ, K * φ, and φφ. The data used in this analysis consist of 2.6 million BB pairs produced at the Υ(4S) taken with the CLEO-II detector at the Cornell Electron Storage Ring (CESR). We measure the branching fraction of the sum of B 0 → π + π − and B 0 → K + π − to be (1.8 +0.6+0.2 −0.5−0.3 ± 0.2) × 10 −5 . In addition, we place upper limits on individual branching fractions in the range from 10 −4 to 10 −6 .