(abbreviated) We aim to study the inner wind of the well-known AGB star CW Leo. Different diagnostics probing different geometrical scales have pointed toward a non-homogeneous mass-loss process: dust clumps are observed at milli-arcsec scale, a bipolar structure is seen at arcsecond-scale and multi-concentric shells are detected beyond 1". We present the first ALMA Cycle 0 band 9 data around 650 GHz. The full-resolution data have a spatial resolution of 0".42x0".24, allowing us to study the morpho-kinematical structure within ~6". Results: We have detected 25 molecular lines. The emission of all but one line is spatially resolved. The dust and molecular lines are centered around the continuum peak position. The dust emission has an asymmetric distribution with a central peak flux density of ~2 Jy. The molecular emission lines trace different regions in the wind acceleration region and suggest that the wind velocity increases rapidly from about 5 R* almost reaching the terminal velocity at ~11 R*. The channel maps for the brighter lines show a complex structure; specifically for the 13CO J=6-5 line different arcs are detected within the first few arcseconds. The curved structure present in the PV map of the 13CO J=6-5 line can be explained by a spiral structure in the inner wind, probably induced by a binary companion. From modeling the ALMA data, we deduce that the potential orbital axis for the binary system lies at a position angle of ~10-20 deg to the North-East and that the spiral structure is seen almost edge-on. We infer an orbital period of 55 yr and a binary separation of 25 au (or ~8.2 R*). We tentatively estimate that the companion is an unevolved low-mass main-sequence star. The ALMA data hence provide us for the first time with the crucial kinematical link between the dust clumps seen at milli-arcsecond scale and the almost concentric arcs seen at arcsecond scale.