We show that asymptotic normalization coefficients (ANC) can be extracted from one nucleon breakup reactions of loosely bound nuclei at 30-300 MeV/u. In particular, the breakup of ^8B is described in terms of an extended Glauber model. The 8B ANC extracted for the ground state of this nucleus from breakup data at several energies and on different targets, C^2 = 0.450+/-0.039 fm^-1, leads to the astrophysical factor S_17(0)= 17.4+/-1.5 eVb for the key reaction for solar neutrino production

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... gamma)8B. The procedure described here is more general, providing an indirect method to determine reaction rates of astrophysical interest with beams of loosely bound radioactive nuclei.