  MAXI J1305-704 has been proposed as a high-inclination candidate black hole
X-ray binary in view of its X-ray properties and dipping behaviour during
outburst. We present photometric and spectroscopic observations of the source
in quiescence that allow us to reveal the ellipsoidal modulation of the
companion star and absorption features consistent with those of an early K-type
star (Teff = 4610 +130 -160 K). The central wavelengths of the absorption lines
vary periodically at Porb = 0.394 +- 0.004 d with an amplitude of K2 = 554 +- 8
km/s . They imply a mass function for the compact object of f(M1) = 6.9 +- 0.3
Msun, confirming its black hole nature. The simultaneous absence of X-ray
eclipses and the presence of dips set a conservative range of allowed
inclinations 60 deg < i < 82 deg, while modelling of optical light curves
further constrain it to i = 72 +5 -8 deg. The above parameters together set a
black hole mass of M1 = 8.9 +1.6 -1.0 Msun and a companion mass of M2=0.43 +-
0.16 Msun, much lower than that of a dwarf star of the observed spectral type,
implying it is evolved. Estimates of the distance to the system (d = 7.5 +1.8
-1.4 kpc) and space velocity (vspace = 270 +- 60 km/s ) place it in the
Galactic thick disc and favour a strong natal kick during the formation of the
BH if the supernova occurred in the Galactic Plane.
