Thick, 40 microm plastic shells filled with 25-35 atm of D2 or D3He were imploded on a low-adiabat (alpha approximately 1.3) and with a low-implosion velocity ( approximately 2 x 10(70 cm/s) on the OMEGA laser to generate massive cores of compressed plasma with high areal densities optimal for fast ignition. The targets are driven by 20-kJ relaxation adiabat-shaping laser pulses to keep the inner portion of the shell nearly Fermi degenerate. The measured kinetic energy downshift of proton spectra is in good agreement with the theoretical predictions yielding burn-averaged areal densities of 0.130+/-0.017 g/cm2 and peak rhoR during the burn of about 0.24+/-0.018 g/cm2, the largest rhoR measured on OMEGA to date. The same implosions with empty plastic shells are expected to reach 1.3 g/cm2 across the core (i.e., 2rhoR) enough to stop fast electrons with energies up to 4.5 MeV typical of fast ignition scenarios.