The good energy resolution (3--4 keV FWHM) of the Transient Gamma Ray Spectrometer (TGRS) on board the WIND spacecraft makes it sensitive to Doppler-shifted outbursts of 511 keV electron-positron annihilation radiation, the reason being that the Doppler shift causes the cosmic line to be slightly offset from a strong instrumental background 511 keV line at rest, which is ubiquitous in space environments. Such a cosmic line (blueshifted) is predicted to arise in classical novae due to the annihilation of positrons from $beta$-decay on a timescale of a few hours in an expanding envelope. A further advantage of TGRS - its broad field of view, containing the entire southern ecliptic hemisphere - has enabled us to make a virtually complete and unbiased 3-year search for classical novae at distances up to ~1 kpc. We present negative results of this search, and estimate its implications for the highly-uncertain Galactic classical nova rate and for future space missions.