We present a direct measurement of the low-energy 8Li(p ,α)5He cross section, using a radioactive 8Li beam impinging on a thick target. With four beam energies, we cover the energy range between Ec.m.=0.2 and 2.1 MeV. An R-matrix analysis of the data is performed and suggests the existence of two broad overlapping resonances (5/2+ at Ec.m.=1.69 MeV and 7/2+ at E c.m.=1.76 MeV). At low energies our data are sensitive to the properties of a subthreshold state (Ex=16.67 MeV) and of two resonances above threshold. These resonances were observed in previous experiments. The R-matrix fit confirms spin assignments, and provides partial widths. We propose a new 8Li(p,α)5He reaction rate and briefly discuss its influence in nuclear astrophysics. © 2012 American Physical Society. ; SCOPUS: ar.j ; info:eu-repo/semantics/published