A novel multilayer mirror was designed and fabricated based on a recently developed three-material technology aimed both at reaching reflectivities of about 20% and at controlling dispersion over a bandwidth covering photon energies between 35 and 50 eV. The spectral phase upon reflection was retrieved by measuring interferences in a two-color ionization process using high-order harmonics produced from a titanium: sapphire laser. We demonstrate the feasibility of designing and characterizing phase-controlled broadband optics in the extreme-ultraviolet domain, which should facilitate the manipulation of attosecond pulses for applications.