  We place direct observational constraints on the black-hole masses of the
cosmologically important z~2 submillimeter-emitting galaxy (SMG; f850>4mJy)
population, and use measured host-galaxy masses to explore their evolutionary
status. We employ the well-established virial black-hole mass estimator to
'weigh' the black holes of a sample of z~2 SMGs with broad Halpha or Hbeta
emission. The average black-hole mass and Eddington ratio (eta) of the
lower-luminosity broad-line SMGs (L_X~10^44 erg/s} are log(M_BH/M_sol)~8.0 and
eta~0.2, respectively. These lower-luminosity broad-line SMGs lie in the same
location of the L_X-L_FIR plane as more typical SMGs hosting X-ray obscured AGN
and may be intrinsically similar systems, but orientated so that the rest-frame
optical nucleus is visible. Under this hypothesis, we conclude that SMGs host
black holes with log(M_BH/M_odot)~7.8; we find supporting evidence from
observations of local ULIRGs. Combining these black-hole mass constraints with
measured host-galaxy masses, we find that the black holes in SMGs are >3 times
smaller than those found in comparably massive normal galaxies in the local
Universe, albeit with considerable uncertainty, and >10 times smaller than
those predicted for z~2 luminous quasars and radio galaxies. These results
imply that the growth of the black hole lags that of the host galaxy in SMGs,
in stark contrast with that previously suggested for radio galaxies and
luminous quasars at z~2. On the basis of current host-galaxy mass constraints,
we show that SMGs and their descendants cannot lie significantly above the
locally defined M_BH-M_GAL relationship. We argue that the black holes in the
z~0 descendents of SMGs will have log(M_BH/M_odot)~8.6, indicating that they
only need to grow by a factor of ~6 by the present day (ABRIDGED).
