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 5 \times 10 ^ { 10 } ~ { } M _ { \odot } of molecular gas within 10 kpc of the BCG . Its ensemble velocity profile width of \sim 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 10 ^ { 10 } ~ { } M _ { \odot } 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 ^ { -1 } to +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 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 a hot outflow on larger scales reported by Kirkpatrick and colleagues . The molecular gas flow rate of approximately 200 ~ { } M _ { \odot } ~ { } yr ^ { -1 } is comparable to the star formation rate of 100 - 180 ~ { } M _ { \odot } ~ { } 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 , it is able to sweep higher density molecular gas away from their centers .