  We propose a method for amplifying a train of sub-femtosecond pulses of
circularly or elliptically polarized extreme ultraviolet (XUV) radiation
constituted by high-order harmonics of an infrared (IR) laser field, in a
neon-like active medium of a plasma-based X-ray laser, additionally irradiated
with a replica of a fundamental frequency IR field. It is shown that the
ellipticity of the pulses can be maintained or increased during the
amplification process. The experimental implementation is suggested in an
active medium of an X-ray laser based on neon-like Ti^{12+} ions irradiated by
an IR laser field with a wavelength of 3.9 microns.
