Doppler tracking of deep space probes is central to spacecraft navigation and many radio science investigations. The most sensitive Doppler observations to date were taken using the NASA/JPL Deep Space Network antenna DSS 25 (a 34 m diameter beam-waveguide station instrumented with simultaneous X- and Ka-band uplink and tropospheric scintillation calibration equipment) tracking the Cassini spacecraft. Those observations achieved Doppler fractional frequency stability (Doppler frequency fluctuation divided by center frequency, Δf/fo) ≈ 3 X 10−15 at 1000 s integration time. The leading noise in these very-high-sensitivity tracks was time-dependent unmodeled motion of the ground antenna's phase center (caused, e.g., by antenna sag as elevation angle changes, unmodeled subreflector motion, wind loading, bulk motion of the antenna as it rolled over irregularities in its azimuth ring, etc.). This antenna mechanical noise has seemed irreducible since it is not clear how to build a large, moving, steel structure with intrinsic mechanical stability better than that of current tracking stations. Here we show how intrinsic mechanical noise of a large tracking antenna can be suppressed when two-way Doppler tracking data and receive-only Doppler data from a stiffer antenna are combined with suitable delays. Using this time delay correction procedure, the mechanical noise in the final Doppler observable can be reduced to that of the stiffer antenna. We demonstrate proof-of-concept experimentally and briefly discuss some practical considerations.