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 SgrA* whose infrared counterpart may have been detected, for the first time, in a deep image in 1996 June. We also report lambda/Deltalambda~35 speckle spectroscopy and show that several of the SgrA* (infrared) cluster members are likely early-type stars of mass ~15 to 20 Msolar. 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 (>=10^3 km s^-1) in the immediate vicinity (0.01 pc) of SgrA*. From the stellar radial and proper motion data, we infer that a dark mass of 2.61 (+/-0.15_stat)(+/-0.35_stat+sys)x10^6 Msolar must reside within about one light-week of the compact radio source. Its density must be 2.2x10^12 Msolar 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 (>=10^5 Msolar) is associated with the compact radio source SgrA* itself and is concentrated on a scale of less than 15 times the Schwarzschild radius of a 2.6x10^6-Msolar black hole. The corresponding density is 3x10^20 Msolar 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.