The S=1/2 spin chain material SrCuO2 doped with 1% S=1 Ni impurities is studied by inelastic neutron scattering. At low temperatures, the spectrum shows a pseudogap Δ≈8  meV, absent in the parent compound, and not related to any structural phase transition. The pseudogap is shown to be a generic feature of quantum spin chains with dilute defects. A simple model based on this idea quantitatively accounts for the experimental data measured in the temperature range from 2 to 300 K, and allows us to represent the momentum-integrated dynamic structure factor in a universal scaling form.