The low mass X-ray binary Aquila X-1 is one of the most active neutron star X-ray transients. Despite it has a relatively bright quiescent optical counterpart, the detection of its companion has been hampered by the presence of a nearby interloper star. Using the infrared integral field spectrograph SINFONI on the VLT-8.2m telescope, we unambiguously single out Aquila X-1 from the interloper. Phase-resolved near infrared spectroscopy reveals absorption features from a K4 +- 2 companion star moving at a projected velocity of K_2= 136 +- 4 km/s. We here present the first dynamical solution and associated fundamental parameters of Aquila X-1, imposing new constraints to the orbital inclination (36 deg < i < 47 deg) and the distance (d = 6 +- 2 kpc) to this prototypical neutron star transient.