High-accuracy lattice sums have been evaluated for the Lennard-Jones 12-6 pair potential, without cutoffs, in the close-packed fcc and hcp lattices. The results confirm the small relative stability of hcp at low pressure, and locate precisely the first-order phase transition at zero temperature to the fcc structure. The reduced pressure p sigma (3)/epsilon at this transition is approximately 878.476... , with both structures having been compressed to about one-half of their zero-pressure volumes. On account of its lower symmetry compared to fcc, the hcp lattice spontaneously distorts from the ideal close-packed geometry to lower its energy by a tiny amount. For low compressions, this distortion involves expansion within close-packed planes, and shrinkage in the perpendicular stacking direction. However this spontaneous distortion changes sign shortly before reaching the compression required for the hcp-fcc phase transition, vanishing at a volume ratio (compared to zero pressure) of about 0.537.