The nuclear physics input from the 3He(alpha,gamma)7Be cross section is a major uncertainty in the fluxes of 7Be and 8B neutrinos from the Sun predicted by solar models and in the 7Li abundance obtained in big-bang nucleosynthesis calculations. The present work reports on a new precision experiment using the activation technique at energies directly relevant to big-bang nucleosynthesis. Previously such low energies had been reached experimentally only by the prompt-gamma technique and with inferior precision. Using a windowless gas target, high beam intensity, and low background gamma-counting facilities, the 3He(alpha,gamma)7Be cross section has been determined at 127, 148, and 169 keV center-of-mass energy with a total uncertainty of 4%. The sources of systematic uncertainty are discussed in detail. The present data can be used in big-bang nucleosynthesis calculations and to constrain the extrapolation of the 3He(alpha,gamma)7Be astrophysical S factor to solar energies.