We report ALMA Early Science observations of the A1835 brightest cluster galaxy (BCG) in the CO (3-2) and CO (1-0) emission lines. We detect 5 × 1010 M ☉ of molecular gas within 10 kpc of the BCG. Its ensemble velocity profile width of ~130 km s–1 FWHM is too narrow for the molecular clouds to be supported in the galaxy by dynamic pressure. The gas may instead be supported in a rotating, turbulent disk oriented nearly face-on. Roughly 1010 M ☉ of molecular gas is projected 3-10 kpc to the northwest and to the east of the nucleus with line-of-sight velocities lying between –250 km s–1 and +480 km s–1 with respect to the systemic velocity. The high-velocity gas may be either inflowing or outflowing. However, the absence of high-velocity gas toward the nucleus that would be expected in a steady inflow, and its bipolar distribution on either side of the nucleus, are more naturally explained as outflow. Star formation and radiation from the active galactic nucleus (AGN) are both incapable of driving an outflow of this magnitude. The location of the high-velocity gas projected behind buoyantly rising X-ray cavities and favorable energetics suggest an outflow driven by the radio AGN. If so, the molecular outflow may be associated with a hot outflow on larger scales reported by Kirkpatrick and colleagues. The molecular gas flow rate of approximately 200 M ☉ yr–1 is comparable to the star formation rate of 100-180 M ☉ yr–1 in the central disk. How radio bubbles would 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, but it is able to sweep higher density molecular gas away from their centers.