  We study ferroelectricity in the classic perovskite ferroelectric PbTiO$_3$
to high pressures with density functional theory (DFT) and experimental
diamond-anvil techniques. We use second harmonic generation (SHG) spectroscopy
to detect lack of inversion symmetry, if present. Consistent with early
understanding and experiments, we find that ferroelectricity disappears at
moderate pressures. However, the DFT calculations show that the disappearance
arises from the overtaking of zone boundary instabilities, and not from the
squeezing out of the off-centering ferroelectric displacements with pressure,
as previously thought. We find a new dense centrosymmetric post-perovskite
phase with P$2_1/m$ symmetry that becomes stable at about 70 GPa. Our
computations predict a dramatic reentrance of ferroelectricity 90 GPa in the
distorted R3c structure, but this phase is metastable relative to the
post-perovskite phase.
