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We use the Green Bank Telescope to search for $^3He^+$ emission from a sampleof four Galactic planetary nebulae: NGC 3242, NGC 6543, NGC 6826, and NGC 7009.During the era of primordial nucleosynthesis the light elements $^2H$, $^3He$,$^4He$, and $^7Li$ were produced in significant amounts and these abundanceshave since been modified primarily by stars. Observations of $^3He^+$ in H IIregions located throughout the Milky Way disk reveal very little variation inthe $^3He/H$ abundance ratio -- the "$^3He$ Plateau" -- indicating that the neteffect of $^3He$ production in stars is negligible. This is in contrast to muchhigher $^3He/H$ abundance ratios reported for some planetary nebulae. Thisdiscrepancy is known as the "$^3He$ Problem". We use radio recombination linesobserved simultaneously with the $^3He^+$ transition to make a robustassessment of the spectral sensitivity that these observations achieve. Wedetect spectral lines at $\sim$ 1 -- 2 mK intensities, but at these levelsinstrumental effects compromise our ability to measure accurate spectral lineparameters. We do not confirm reports of previous detections of $^3He^+$ in NGC3242 nor do we detect $^3He^+$ emission from any of our sources. This resultcalls into question all reported detections of $^3He^+$ emission from anyplanetary nebula. The $^3He/H$ abundance upper limit we derive here for NGC3242 is inconsistent with standard stellar production of $^3He$ and thusrequires that some type of extra mixing process operates in low-mass stars.

Green Bank Telescope Observations of ${\bf ^3He^{\bf +}}$: Planetary Nebulae