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 ~500 ms and spectral indices of ~-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 (GC) to date. No convincing candidate was detected in the survey, with a 10σ detection threshold of ~10 μJy achieved in two separate observing sessions. This survey represents a significant improvement over previous searches for pulsars at the GC and would have detected a significant fraction (gsim5%) 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*.