We report the first experimental demonstration of a-periodic multilayer mirrors controlling the frequency sweep (chirp) of isolated attosecond XUV pulses. The concept was proven with about 200-attosecond pulses in the photon energy range of 100-130 eV measured via photoelectron streaking in neon. The demonstrated attosecond dispersion control is engineerable in a wide range of XUV photon energies and bandwidths. The resultant tailor-made attosecond pulses with highly enhanced photon flux are expected to significantly advance attosecond metrology and spectroscopy and broaden their range of applications.