The long-term stability of optical parametric chirped-pulse amplifiers is hindered by thermal path length drifts affecting the temporal pump-to-signal overlap. A kilowatt-pumped burst amplifier is presented delivering broadband 1.4 mJ pulses with a spectral bandwidth supporting sub-7 fs pulse duration. Active temporal overlap control can be achieved by feedback of optical timing signals from cross-correlation or spectral measurements. Using a balanced optical cross-correlator, we achieve a pump-to-signal synchronization with a residual jitter of only (46 ± 2) fs rms. Additionally, we propose passive pump-to-signal stabilization with an intrinsic jitter of (7.0 ± 0.5) fs rms using white-light continuum generation.