  We suggest a technique to amplify a train of attosecond pulses, produced by
high-harmonic generation (HHG) of an infrared (IR) laser field, in an active
medium of a plasma-based X-ray laser. This technique is based on modulation of
transition frequency of the X-ray laser by the same IR field, as used to
generate the harmonics, via linear Stark effect, which results in
redistribution of the resonant gain and simultaneous amplification of a wide
set of harmonics in the incident field. We propose an experimental
implementation of the suggested technique in active medium of C5+ ions at
wavelength 3.4 nm in the "water window" range and show the possibility to
amplify by two orders of magnitude a train of attosecond pulses with pulse
duration down to 100 as. We show also a possibility to isolate a single
attosecond pulse from the incident attosecond pulse train during its
amplification in optically deep modulated medium.
