We discuss constraints on the properties and nature of the dark mass concentration at the core of the Milky Way. We present 0.15-arcsec astrometric K-band maps in five epochs beween 1992 and 1996. From these we derive imposed stellar proper motions within 3 arcsec of the compact radio source Sgr A* whose infrared counterpart may have been detected, for the first time, in a deep image in 1996 June. We also report lambda/Delta lambda similar to 35 speckle spectroscopy and show that several of the Sgr A* (infrared) cluster members are likely early-type stars of mass similar to 15 to 20 M.. All available checks, including a first comparison with high-resolution maps that are now becoming available from other groups, support our previous conclusion that there are several fast-moving stars (greater than or equal to 10(3) km s(-1)) in the immediate vicinity (0.01 pc) of Sgr A*. From the stellar radial and proper motion data, we infer that a dark mass of 2.61 (+/-0.15(stat))(+/-0.35(stat+sys)) x 10(6) M. must reside within about one light-week of the compact radio source. Its density must be 2.2 x 10(12) M. pc(-3) or greater. There is no stable configuration of normal stars, stellar remnants or substellar entities at that density. From an equipartition argument we infer that at least 5 per cent of the dark mass (greater than or equal to 10(5) M.) is associated with the compact radio source Sgr A* itself and is concentrated on a scale of less than 15 times the Schwarzschild radius of a 2.6 x 10(6)-M. black hole. The corresponding density is 3 x 10(20) M. pc(-3) or greater. If one accepts these arguments it is hard to escape the conclusion that there must be a massive black hole at the core of the Milky Way.