Cross sections for the 3He(α,γ)7Be reaction have been measured at several energies from Ecm = 165 to 1200 keV by counting prompt γ rays from a windowless, differentially pumped, recirculating, 3He gas target. The cross-section factor S34(Ecm) and branching ratio γ1/γ0 were determined at each energy. Absolute cross sections were measured at Ecm = 947 and 1255 keV by counting the γ rays from the 7Be produced in a 3He gas cell with a Ni entrance foil. The inferred zero-energy intercept of the cross-section factor is S34(0) = 0.52 ± 0.03 keV-barn. This value is in good agreement with the previous measurements of Parker and Kavanagh and Nagatani et al., but disagrees substantially with the recently published measurement of Kräwinkel et al. from Münster. The relationship between the present measurements and several theoretical calculations, and the effect of the extrapolated cross section on the solar neutrino problem are discussed.