We study the phase diagram of an effective ion model of fully ionized hydrogen at ultra-high pressure. We assume that the protons interact with a screened Coulomb potential derived from a static linear response theory. This model accurately reproduces the physical properties of hydrogen for densities greater than ρm=10g/cm3ρm=10g/cm3 corresponding to the range of the coupling parameter rs ≲ 0.6. The pressure range, P≳20TPaP≳20TPa, is well beyond present experimental limitations. Assuming classical protons, we find that the zero temperature enthalpy of the perfect bcc crystal is slightly lower than for other structures at ρm=12.47g/cm3ρm=12.47g/cm3 while the fcc structure gains stability at higher density. Using Monte Carlo calculations, we compute the free energy of various phases and locate the melting transition versus density. We find that on melting, bcc is energetically favored with respect to fcc over the entire range investigated. In the solid phase the system undergoes a structural transition from bcc at higher temperature to fcc at lower temperature. The free energy difference between these two structures is very small so that obtaining a quantitative estimate of this second transition line requires accuracy beyond that provided by our method. We estimate the effect of proton zero point motion on the bcc melting line for hydrogen, deuterium, and tritium by a path integral Monte Carlo method. Although zero point effects on hydrogen are large, since the two competing phases (bcc and liquid) have locally similar environments, the effect on the melting line is small; the melting temperature for hydrogen is lowered by about 10% with respect to the classical value.