We report very large changes under modest hydrostatic pressure in the nature of the relaxor ferroelectric (FE) response for a single crystal of SrTiO3 doped with 0.7 mol % CaTiO3. The Ca cation dopants generate polar clusters within the FE soft mode, readily polarized host crystal. At ambient pressure the cluster size increases with decreasing temperature, leading to a dispersive (relaxor) state below ∼18 K. However, the application of modest pressure stiffens the soft mode frequency of the host lattice and reduces cluster growth, thereby decreasing the glass-like transition temperature at roughly –35 K/kbar. Above 0.5 kbar there is no evidence for the relaxor state; rather, a temperature- and frequency-independent dielectric response reflecting quantum paraelectric behavior evolves. These results demonstrate the extreme sensitivity to pressure of relaxors with low transition temperatures, i.e., in the quantum regime where characteristic energies are small.