Two laser strainmeters that operate at 1400-m depth, about 20 km NE of the epicenter of the 6 April 2009 magnitude-6.3 L'Aquila, Italy, earthquake, have produced a clear record of postseismic strain. Here we show the results from the analysis of the data related to the first few days after the event. Strain after about 1.5 days is fully consistent with afterslip on a stationary region of the earthquake causative fault. The preceding few-hour-long transient (whose seismic moment history is quasi-exponential) is fully consistent with unilateral diffusive slip propagation toward the shallower part of the same fault. The propagation path ends where later afterslip probably occurred. Slip propagation similar to heat diffusion has been suggested to explain the observed scaling law between amplitude and duration of slow earthquakes; here we give the first observational evidence of the role and details of slow rupture propagation.