We report ALMA Early Science observations of the Abell 1835 brightest cluster galaxy (BCG) in the CO (3-2) and CO (1-0) emission lines. We detect 5E10 solar masses of molecular gas within 10 kpc of the BCG. Its velocity width of ~130 km/s FWHM is too narrow to be supported by dynamical pressure. The gas may instead be supported in a rotating, turbulent disk oriented nearly face-on. The disk is forming stars at a rate of 100-180 solar masses per year. Roughly 1E10 solar masses of molecular gas is projected 3-10 kpc to the north-west and to the east of the nucleus with line of sight velocities lying between -250 km/s to +480 km/s with respect to the systemic velocity. Although inflow cannot be ruled out, the rising velocity gradient with radius is consistent with a broad, bipolar outflow driven by radio jets or buoyantly rising X-ray cavities. The molecular outflow may be associated with an outflow of hot gas in Abell 1835 seen on larger scales. Molecular gas is flowing out of the BCG at a rate of approximately 200 solar masses per year, which is comparable to its star formation rate. How radio bubbles lift dense molecular gas in their updrafts, how much gas will be lost to the BCG, and how much will return to fuel future star formation and AGN activity are poorly understood. Our results imply that radio-mechanical (radio mode) feedback not only heats hot atmospheres surrounding elliptical galaxies and BCGs, it is able to sweep higher density molecular gas away from their centers.