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First results are reported for a program of monitoring symbiotic-star velocities in the 1.6 μm region with infrared-array technology. Infrared radial velocities have been used to determine single-lined spectroscopic orbits for six well-known symbiotic stars, EG And, T CrB, CI Cyg, BX Mon, RS Oph, and AG Peg. The new orbits are in general agreement with previous orbits derived from optical velocities. From the combined optical and infrared velocities improved orbital elements for the six systems have been determined. Each of the orbital periods has been determined solely from the radial-velocity data. With the addition of our new velocities, the orbital period of BX Mon has been revised to 1259 days, a 10% decrease from the previously reported result.