The polarization of the X-rays reflected by a centrally illuminated, optically thick and geometrically thin accretion disc is calculated as a function of the source and disc parameters. The degree of polarization of the reflected radiation depends strongly on the disc inclination, and could be greater than 20 percent; the polarization angle is 90 deg, i.e. the polarization vector lies in the meridian plane. Due to the dilution of the primary source, assumed unpolarized, the net polarization depends strongly on photon energy, and could reach values as high as about 6 percent at energies greater than a few tens of keV. When relativistic effects in the Schwarzschild metric are taken into account, the degree of polarization generally increases, and the polarization angle becomes greater than 90 deg. The model could be tested by the SXRP experiment for several galactic X-ray binaries, and at least for one active galaxy, i.e. Centaurus A.