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We report results from a deep high-frequency search for pulsars within the central parsec of Sgr A* using the Green Bank Telescope. The observing frequency of 15 GHz was chosen to maximize the likelihood of detecting normal pulsars (i.e. with periods of $\sim 500$\,ms and spectral indices of $\sim -1.7$) close to Sgr A*, that might be used as probes of gravity in the strong-field regime; this is the highest frequency used for such pulsar searches of the Galactic Center to date. No convincing candidate was detected in the survey, with a $10\sigma$ detection threshold of $\sim 10 \mu$Jy achieved in two separate observing sessions. This survey represents a significant improvement over previous searches for pulsars at the Galactic Center and would have detected a significant fraction ($\gtrsim 5%) of the pulsars around Sgr A*, if they had properties similar to those of the known population. Using our best current knowledge of the properties of the Galactic pulsar population and the scattering material toward Sgr A*, we estimate an upper limit of 90 normal pulsars in orbit within the central parsec of Sgr A*.