The melting properties and thermodynamic functions of solid helium have been determined at temperatures from 4 to 26 degrees K and at pressures up to 3000 atm. The upper temperature corresponds to about five times the critical temperature of helium; it was therefore possible to measure properties of the solid state in a range which has not yet been attained for any other substance. The melting curve shows no signs of an approach to a solid-fluid critical point; in fact, the difference between the phases becomes more pronounced at higher melting temperatures. The internal energy at 0 degrees K was calculated from the experimental data and was found to be in good agreement with the theoretical values based on the Slater-Kirkwood potential, using 9/8Rθ as an estimate of the zero-point energy (θ being the Debye characteristic temperature). A first-order transition in the solid was revealed; its equilibrium line cuts the melting curve at 14\cdot 9 degrees K and moves to higher temperatures at higher densities. The heat of transition is very small, about 0\cdot 08 cal/mole. The transition is assumed to correspond to a change of crystal structure from hexagonal to cubic close-packed. At the highest pressure solid helium is compressed to less than half its volume under equilibrium conditions at absolute zero, and the Debye θ is increased five times. It was hence possible to test the Lindemann melting formula for a single substance over a very wide range. The formula was found to fit the experimental data satisfactorily, although the value of the constant in it differed somewhat from the classical value.