The THz Kerr effect measures the birefringence induced in an otherwise isotropic material by a strong THz pulse driving the Raman-active excitations of the systems. Here we provide experimental evidence of a sizable Kerr response in insulating SrTiO₃ due to infrared-active lattice vibrations. Such a signal, named the ionic Kerr effect, is associated with the simultaneous excitation of multiple phonons. Thanks to a theoretical modeling of the time, polarization, and temperature dependence of the birefrengence, we can disentangle the ionic Kerr effect from the off-resonant electronic excitations, providing an alternative tunable mechanism to modulate the refractive index on ultrashort timescales via infrared active phonons.