A superconducting accelerator-pumped free-electron laser (SCA/FEL) is used to perform photon echo experiments on a near-IR dye molecule in a polymeric glass. The SCA/FEL produces Fourier-transform-limited picosecond pulses at a repetition rate of 11.8 MHz. Its wavelength is continuously tunable. The experiments are conducted at 0.776-mu-m by doubling the output of the SCA/FEL tuned to 1.55-mu-m. At this wavelength, the SCA/FEL is operated in a burst mode. The duration of the bursts (macropulses) is 2 ms with a repetition rate of 10 Hz. Each individual pulse (micropulse) has a length of 3.2 ps and is approximately 1-mu-J in energy. In the experiment a number of single micropulses are selected from each macropulse. The doubling, stabilization, synchronization, single-pulse selection, and detection electronics are described. These are the first photon echo experiments and, to our knowledge, the first optical coherence experiments performed using a free-electron laser as a source.