The polytropic equation of state of an atomic hydrogen gas is examined for primordial halos with baryonic masses of M_h ~ 10⁷-10⁹ M. For roughly isothermal collapse around 10⁴ K, we find that line trapping of Lyα (H I and He II) photons causes the polytropic exponent to stiffen to values significantly above unity. Under the assumptions of zero H₂ abundance and very modest pollution by metals (<10^-4 solar), fragmentation is likely to be inhibited for such an equation of state. We argue on purely thermodynamic grounds that a single black hole of ~(0.02-0.003)M_h can form at the center of a halo for z = 10-20 when the free-fall time is less than the time needed for a resonantly scattered Lyα photon to escape from the halo. The absence of H₂ follows naturally from the high temperatures, >10⁴ K, that are attained when Lyα photons are trapped in the dense and massive halos that we consider. An H₂-dissociating UV background is needed if positive feedback effects on H₂ formation from X-rays occur. The black hole-to-baryon mass fraction is suggestively close to what is required for these intermediate-mass black holes, of mass M_BH ~ 10⁴-10⁶ M, to act as seeds for forming the supermassive black holes of mass ~0.001M_spheroid found in galaxies today.