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We demonstrate all-optical storage of a picosecond-pulse packet of ones and zeros in a fiber buffer in which loss is compensated by parametric amplification. An all-fiber phase-sensitive parametric amplifier, which exploits a nonlinear Sagnac loop made of standard polarization-maintaining fiber to provide gain, regenerates the stored pulses on each round-trip. Storage for periods of time tip to 1 ms has been observed. Data packets of 10's of kilobits could be stored in this device with use of a higher repetition-rate pump source.