Field-dependent microwave absorption in high-quality YBa2Cu3O7 single crystals shows that even very low magnetic fields penetrate into the sample. In the temperature interval of a few degrees below Tc, a broad hysteretic signal is observed. We interpret it in terms of a dense structure of weak links, whose origin is in twin boundaries and other thin defect planes. As the temperature is reduced, the critical currents of these junctions increase rapidly, and approach the bulk value. The broad signal then vanishes. At low temperatures one can observe narrow discrete absorption lines due to occasional defects of a different kind, which are thicker and, therefore, maintain a weak coupling. The dense structure of weak links, active just below Tc, may preclude the measurements of some intrinsic bulk superconducting properties close to Tc, and invalidate a straightforward comparison of experimental data with theoretical predictions.