  The effects of high magnetic fields applied perpendicular to the spontaneous
ferroelectric polarization on single crystals of BiFeO$_3$ were investigated
through magnetization, magnetostriction, and neutron diffraction measurements.
The magnetostriction measurements revealed lattice distortion of $2\times
10^{-5}$, during the reorientation process of the cycloidal spin order by
applied magnetic fields. Furthermore, anomalous changes in magnetostriction and
electric polarization at a larger field demonstrate an intermediate phase
between cycloidal and canted antiferromagnetic states, where a large
magnetoelectric effect was observed. Neutron diffraction measurements clarified
that incommensurate spin modulation along [110] direction in the cycloidal
phase becomes commensurate in the intermediate phase. Theoretical calculations
based on the standard spin Hamiltonian of this material suggest an
antiferromagnetic cone-type spin order in the intermediate phase.
