We report R 6 4300 VLT SINFONI adaptive optics integral field K-band spectroscopy of the nucleus of the Seyfert 1 galaxy NGC 3227 at a spatial resolution of 0".085 (7 pc). We present the morphologies and kinematics of emission lines and absorption features and give the first derivation of a black hole mass in a Seyfert 1 nucleus from stellar dynamics (marginally resolving the black hole's sphere of influence). We show that the gas in the nucleus has a mean column density of order 10 super(24) cm super(-2) and that it is geometrically thick, in agreement with the standard "molecular torus" scenario. We discuss possible heating processes responsible for maintaining the vertical height of the torus. We also resolve the nuclear stellar distribution and find that within a few parsecs of the AGN there has been an intense starburst, the most recent episode of which began 640 Myr ago but has now ceased. The current luminosity of stars within 30 pc of the AGN, 63 x 10 super(9) L sub( ), is comparable to that of the AGN. We argue that the star formation has been occurring in the obscuring material. Finally, we apply Schwarzschild orbit superposition models to our full two-dimensional data and derive the mass of the black hole, paying careful attention to the input parameters, which are often uncertain. Our models yield a 1 s range for the black hole mass of M sub(BH) = 7 x 10 super(6)-2 x 10 super(7) M sub( ).