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The mechanical and electrical quality factors of a 10‐g niobium resonator were measured at 4.4 K and were found to be 8.1×106, and 3.8×106, respectively. The value for the electrical quality factor is high enough for a system operating at 50 mK at a sensitivity level of one phonon. The resonator’s low damping properties make it suitable for use as a transducer for a cryogenic three‐mode gravitational radiation detector. A practical design is given for the mounting of the resonator on a 2400‐kg aluminum‐bar detector. Projections are made for the sensitivity of a 2400‐kg bar instrumented as a three‐mode system with this resonator inductively coupled to a SQUID.