Synchrotron x-ray diffraction experiment shows that the metal-insulator transition occurring in a ferromagnetic state of a hollandite is accompanied by a structural distortion from the tetragonal to monoclinic phase with a supercell. Detailed electronic structure calculations demonstrate that the metal-insulator transition is caused by a Peierls instability in the quasi-one-dimensional column structure made of four coupled Cr-O chains running in the direction, leading to the formation of tetramers of Cr ions below the transition temperature. This provides a rare example of the Peierls transition of fully spin-polarized electron systems.