  Based on the first-principles electronic structure calculations, we predict
that a chromium oxide K$_2$Cr$_8$O$_{16}$ of hollandite type should be a
half-metallic ferromagnet where the Fermi level crosses only the majority-spin
band, whereas the minority-spin band has a semiconducting gap. We show that the
double-exchange mechanism is responsible for the observed saturated
ferromagnetism. We discuss possible scenarios of the metal-insulator transition
observed at low temperature and we argue that the formation of the
incommensurate, long-wavelength density wave of spinless fermions caused by the
Fermi-surface nesting may be the origin of the opening of the charge gap.
