We present Atacama large millimeter/Â submillimeter array ( ALMA ) and compact array ( ACA ) [ CI ] ^ { 3 } P _ { 1 } - ^ { 3 } P _ { 0 } ( [ CI ] ( 1-0 ) ) observations of NGCÂ 6240 , which we combine with ALMA CO ( 2-1 ) Â and IRAM Plateau de Bure Interferometer CO ( 1-0 ) Â data to study the physical properties of the massive molecular ( H _ { 2 } ) outflow . We discover that the receding and approaching sides of the H _ { 2 } outflow , aligned east-west , exceed 10Â kpc in their total extent . High resolution ( 0.24 \arcsec ) [ CI ] ( 1-0 ) Â line images surprisingly reveal that the outflow emission peaks between the two active galactic nuclei ( AGN ) , rather than on either of the two , and that it dominates the velocity field in this nuclear region . We combine the [ CI ] ( 1-0 ) Â and CO ( 1-0 ) Â data to constrain the CO-to-H _ { 2 } conversion factor ( \alpha _ { CO } ) in the outflow , which is on average 2.1 \pm 1.2 ~ { } M _ { \odot } ( K~ { } km~ { } s ^ { -1 } ~ { } pc ^ { 2 } ) ^ { -1 } . We estimate that 60 \pm 20 Â % of the total H _ { 2 } gas reservoir of NGCÂ 6240 is entrained in the outflow , for a resulting mass-loss rate of \dot { M } _ { out } = 2500 \pm 1200 ~ { } M _ { \odot } ~ { } yr ^ { -1 } \equiv 50 \pm 30 Â SFR . This energetics rules out a solely star formation-driven wind , but the puzzling morphology challenges a classic radiative-mode AGN feedback scenario . For the quiescent gas we compute \langle \alpha _ { CO } \rangle = 3.2 \pm 1.8 ~ { } M _ { \odot } ( K~ { } km~ { } s ^ { -1 } ~ { } pc ^ % { 2 } ) ^ { -1 } , which is at least twice the value commonly employed for ( ultra ) luminous infrared galaxies ( ( U ) LIRGs ) . We observe a tentative trend of increasing r _ { 21 } \equiv L ^ { \prime } _ { CO ( 2 - 1 ) } / L ^ { \prime } _ { CO ( 1 - 0 ) } ratios with velocity dispersion and measure r _ { 21 } > 1 in the outflow , whereas r _ { 21 } \simeq 1 in the quiescent gas . We propose that molecular outflows are the location of the warmer , strongly unbound phase that partially reduces the opacity of the CO lines in ( U ) LIRGs , hence driving down their global \alpha _ { CO } Â and increasing their r _ { 21 } values .