We have studied the high-pressure iron bcc to hcp phase transition by simultaneous x-ray magnetic circular dichroism and x-ray absorption spectroscopy with an x-ray energy dispersive spectrometer. The combination of the two techniques allows us to obtain simultaneously information on both the structure and the magnetic state of iron under pressure. The magnetic and structural transitions simultaneously observed are sharp. Both are of first order in agreement with the theoretical prediction. The pressure domain of the transition observed (2.4+/-0.2 GPa) is narrower than that usually cited in the literature (8 GPa). Our data indicate that the magnetic transition slightly precedes the structural one, suggesting that the origin of the instability of the bcc phase in iron with increasing pressure is to be attributed to the effect of pressure on magnetism as predicted by spin-polarized full-potential total energy calculations.