High-quality and impurity-free magnetite surfaces with (sqrt2xsqrt2)R45o reconstruction have been obtained for the Fe3O4(001) epitaxial films deposited on Fe(001). Based on atomically resolved STM images for both negative and positive sample polarity and Density Functional Theory calculations, a model of the magnetite (001) surface terminated with Fe ions forming dimers on the reconstructed (sqrt2xsqrt2)R45o octahedral iron layer is proposed.