The response of a bulk dielectric to an intense few-cycle laser pulse is not solely determined by the pulse envelope, but also by ultrafast processes occurring during each optical cycle. Here, a method is presented for measuring the retardation of a probe pulse in a strong-field pumped, bulk dielectric with subcycle resolution in the pump-probe delay. Comparisons to model calculations show that the measurement is sensitive to the timing of the electronic Kerr response. When conduction band states are transiently populated at the crests of the laser field, the measurement is also sensitive to the interband dephasing time.