The inferred density of the solid inner core indicates that it is predominantly made of iron. In order to interpret the observed seismic anisotropy and understand the high pressure and temperature behavior of the core, it is essential to establish the crystal structure of iron under core conditions. On the basis of extrapolated experimental data, a number of candidate structures for the high P/T iron phase have been proposed, namely, body-centred cubic (bcc), body-centred tetragonal (bct), hexagonal close-packed (hcp), double-hexagonal close-packed (dhcp) and an orthorhombically distorted hcp polymorph (Matsui, 1993; Stixrude and Cohen, 1995; Boehler, 1993; Saxena et al., 1996; Andrault et al., 1997). Here we present the results of the first fully ab initio free energy calculations for all of these polymorphs of iron at core pressures and temperatures. Our results show that hcp-Fe is the most stable polymorph of iron under the conditions of the Earth's inner core. ¿ 1999 American Geophysical Union