Trimethylamine-N-oxide (TMAO) and urea represent the extremes among the naturally occurring organic osmolytes in terms of their ability to stabilize/destabilize proteins. Their mixtures are found in nature and have generated interest in terms of both their physiological role and their potential use as additives in various applications (crystallography, drug formulation, etc.). Here we report experimental density and activity coefficient data for aqueous mixtures of TMAO with urea. From these data we derive the thermodynamics and solvation properties of the osmolytes, using Kirkwood-Buff theory. Strong hydrogen-bonding at the TMAO oxygen, combined with volume exclusion, accounts for the thermodynamics and solvation of TMAO in aqueous urea. As a result, TMAO behaves in a manner that is surprisingly similar to that of hard-spheres. There are two mandatory solvation sites. In plain water, these sites are occupied with water molecules, which are seamlessly replaced by urea, in proportion to its volume fraction. We discuss how this result gives an explanation both for the exceptionally strong exclusion of TMAO from peptide groups and for the experimentally observed synergy between urea and TMAO.