LTT 9779 b is an ultra-hot Neptune (Rp ≍ 4.7 R⊕, Mp ≍ 29 M⊕) orbiting its Sun-like host star in just 19 hr, placing it deep within the "hot-Neptune desert," where Neptunian planets are seldom found. We present new James Webb Space Telescope NIRSpec G395H phase curve observations that probe its atmospheric composition in unprecedented detail. At near-IR wavelengths, which penetrate the high-altitude clouds inferred from previous NIRISS/SOSS spectra, thermal emission reveals a carbon-rich atmosphere with opacity dominated by carbon monoxide (CO) and carbon dioxide (CO2). Evidence for both species is observed at all orbital phases (σ < 11.6), with retrieved mixing ratios of ∼10−1 for CO and ∼10−4 for CO2, indicating a globally well-mixed reservoir of carbon-bearing gases. We also find moderate evidence of water vapor (H2O; σ < 3.9) on the eastern dayside and dayside (orbital phases 120° and 180°, respectively), and tentatively detect sulfur dioxide (SO2; σ < 2.5) on the western nightside (orbital phase 300°), providing insight into its chemistry and possible photochemical production under intense stellar irradiation. Our findings show that LTT 9779 b retains a carbon-rich atmosphere at a distance from its host star where hot-Neptune-class worlds are expected to evaporate. This makes LTT 9779 b a valuable laboratory for studying atmospheric escape and chemical processes under extreme conditions, offering new insight into the survival of planets in the hot-Neptune desert.