  I report the discovery of a new effect of General Relativity which is
important to understand very rapidly rotating (Kerr) black holes. The orbital
velocity of a test particle is no longer a monotonic function of the orbit
radius when the spin of the black hole is >0.9953, but displays a local
minimum-maximum structure for radii smaller than 1.8 gravitational radii. There
the rate of change of the orbital velocity per radius unit equals the radial
epicyclic frequency and is exactly one third of the polar epicyclic frequency,
suggesting a 3:1 resonant oscillatory motion of the particle. If associated
with the observed quasi-periods - including the most recent 2005/2006 data -
the mass of the supermassive black hole Sgr A* in the centre of the our Galaxy
is determined to 3.3 million solar masses and the spin is 0.99616.
