We evaluate the width of the ω meson in nuclear matter. We consider the free decay mode of the ω into three pions, which is dominated by ρπ decay, and replace the ρ and π propagators by their medium modified ones. We also take into account the quasielastic and inelastic processes induced by a vector-baryon interaction dominated by vector meson exchange, as well as the contributions coming from the ω → KK mechanism with medium modified K,K propagators. We obtain a substantial increase of the ω width in the medium, reaching a value of 129 ± 10 MeV at normal nuclear matter density for an ω at rest, which comes mainly from ωN → ππN, ωN N → πN N processes associated to the dominant ω → ρπ decay mode. The value of the width increases moderately with momentum, reaching values of around 200 MeV at 600 MeV/c.