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 121 ± 10 MeV at normal nuclear matter density for an ω at rest, which comes mainly from ωN → ππN, ωNN → πNN processes associated to the dominant ω → ρπ decay mode. The value of the width increases moderately with momentum, reaching values of around 200MeV at 600MeV/c.