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High-quality dielectric-semiconductor interfaces are critical for reliable high-performance transistors. We report the in-situ metalorganic chemical vapor deposition (MOCVD) of Al$_2$O$_3$ on $\beta$-Ga$_2$O$_3$ as a potentially better alternative to the most commonly used atomic layer deposition (ALD). The growth of Al$_2$O$_3$ is performed in the same reactor as Ga$_2$O$_3$ using trimethylaluminum and O$_2$ as precursors without breaking the vacuum at a growth temperature of 600 $^0$C. The fast and slow near interface traps at the Al$_2$O$_3$/ $\beta$-Ga$_2$O$_3$ interface are identified and quantified using stressed capacitance-voltage (CV) measurements on metal oxide semiconductor capacitor (MOSCAP) structures. The density of shallow and deep level initially filled traps (D$_{it}$) are measured using ultra-violet (UV) assisted CV technique. The average D$_{it}$ for the MOSCAP is determined to be 7.8 $\times$ 10$^{11}$ cm$^{-2}$eV$^{-1}$. The conduction band offset of the Al$_2$O$_3$/ Ga$_2$O$_3$ interface is also determined from CV measurements and found out to be 1.7 eV which is in close agreement with the existing literature reports of ALD Al$_2$O$_3$/ Ga$_2$O$_3$ interface. The current-voltage characteristics are also analyzed and the average breakdown field is extracted to be approximately 5.8 MV/cm. This in-situ Al$_2$O$_3$ dielectric on $\beta$-Ga$_2$O$_3$ with improved dielectric properties can enable Ga$_2$O$_3$-based high performance devices.