This paper determines the thermodynamic phase diagram of the EXP system of particles interacting by the purely repulsive exponential pair potential. The solid phase is face-centered cubic (fcc) at low densities and pressures. At higher densities and pressures, the solid phase is body-centered cubic (bcc) with a re-entrant liquid phase at the highest pressures simulated. The investigation first identifies the phase diagram at zero temperature at which the following four crystal structures are considered: fcc, bcc, hexagonal close packed, and cubic diamond. There is a T = 0 phase transition at pressure 2.651 × 10-3 with the thermodynamically stable structure being fcc below and bcc above this pressure. The densities of the two crystal structures at the phase transition are 1.7469 × 10-2 (fcc) and 1.7471 × 10-2 (bcc). At finite temperatures, the fcc-bcc, fcc-liquid, and bcc-liquid coexistence lines are determined by numerical integration of the Clausius-Clapeyron equation and validated by interface-pinning simulations at selected state points. The bcc-fcc phase transition is a weak first-order transition. The liquid-fcc-bcc triple point, which is determined by the interface-pinning method, has temperature 5.9 × 10-5 and pressure 2.5 × 10-6; the triple-point densities are 1.556 × 10-3 (liquid), 1.583 × 10-3 (bcc), and 1.587 × 10-3 (fcc).