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We report the detection of a transiting, dense Neptune planet candidate orbiting the bright ($V=8.6$) K0.5V star HD 95338. Detection of the 55-day periodic signal comes from the analysis of precision radial velocities from the Planet Finder Spectrograph on the Magellan II Telescope. Follow-up observations with HARPS also confirm the presence of the periodic signal in the combined data. HD 95338 was also observed by the Transiting Exoplanet Survey Satellite ({\it TESS}) where we identify a clear single transit in the photometry. A Markov Chain Monte Carlo period search on the velocities allows strong constraints on the expected transit time, matching well the epoch calculated from \tess{} data, confirming both signals describe the same companion. A joint fit model yields an absolute mass of 42.44$^{+2.22}_{-2.08} M_{\oplus}$ and a radius of 3.89$^{+0.19}_{-0.20}$ $R_{\oplus}$ which translates to a density of 3.98$^{+0.62}_{-0.64}$ \gcm\, for the planet. Given the planet mass and radius, structure models suggest it is composed of a mixture of ammonia, water, and methane. HD 95338\,b is one of the most dense Neptune planets yet detected, indicating a heavy element enrichment of $\sim$90\% ($\sim38\, M_{\oplus}$). This system presents a unique opportunity for future follow-up observations that can further constrain structure models of cool gas giant planets.