We place direct observational constraints on the black-hole masses (M BH) of the cosmologically important z≈ 2 submillimeter-emitting galaxy (SMG; f 850 μm gsim 4 mJy) 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 which exhibit broad Hα or Hβ emission. We find that the average black-hole mass and Eddington ratio (η = L bol/L Edd) of the lower-luminosity broad-line SMGs (L X ≈ 1044 erg s-1) are log(M BH/M sun) ≈ 8.0 and η≈ 0.2, respectively; by comparison, X-ray-luminous broad-line SMGs (L X ≈ 1045 erg s-1) have log(M BH/M sun) ≈ 8.4 and η≈ 0.6. The 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 active galactic nuclei 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 sun) ≈7.8 we find supporting evidence from observations of local ultra-luminous infrared galaxies. Combining these black-hole mass constraints with measured host-galaxy masses, we find that the black holes in SMGs are gsim3 times smaller than those found in comparably massive normal galaxies in the local universe, albeit with considerable uncertainty, and gsim10 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 sun) ≈ 8.6, indicating that they only need to grow by a factor of ≈6 by the present day. We show that this amount of black-hole growth can be achieved within current estimates for the submillimeter-bright lifetime of SMGs, provided that the black holes can grow at rates close to the Eddington limit.