  Tight relationships exist in the local universe between the central stellar
properties of galaxies and the mass of their supermassive black hole. These
suggest galaxies and black holes co-evolve, with the main regulation mechanism
being energetic feedback from accretion onto the black hole during its quasar
phase. A crucial question is how the relationship between black holes and
galaxies evolves with time; a key epoch to probe this relationship is at the
peaks of star formation and black hole growth 8-12 billion years ago (redshifts
1-3). Here we report a dynamical measurement of the mass of the black hole in a
luminous quasar at a redshift of 2, with a look back time of 11 billion years,
by spatially resolving the broad line region. We detect a 40 micro-arcsecond
(0.31 pc) spatial offset between the red and blue photocenters of the H$\alpha$
line that traces the velocity gradient of a rotating broad line region. The
flux and differential phase spectra are well reproduced by a thick, moderately
inclined disk of gas clouds within the sphere of influence of a central black
hole with a mass of 3.2x10$^{8}$ solar masses. Molecular gas data reveal a
dynamical mass for the host galaxy of 6x10$^{11}$ solar masses, which indicates
an under-massive black hole accreting at a super-Eddington rate. This suggests
a host galaxy that grew faster than the supermassive black hole, indicating a
delay between galaxy and black hole formation for some systems.
