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The Compton Spectrometer and Imager (COSI) is a balloon-borne compact Compton telescope designed to survey the 0.2-5 MeV sky. COSI's energy resolution of $\sim$0.2% at 1.8 MeV, single-photon reconstruction, and wide field of view make it capable of studying astrophysical nuclear lines, particularly the 1809 keV $γ$-ray line from decaying Galactic $^{26}$Al. Most $^{26}$Al originates in massive stars and core-collapse supernova nucleosynthesis, but the path from stellar evolution models to Galaxy-wide emission remains unconstrained. In 2016, COSI had a successful 46-day flight on a NASA superpressure balloon. Here, we detail the first search for the 1809 keV $^{26}$Al line in the COSI 2016 balloon flight using a maximum likelihood analysis. We find a Galactic $^{26}$Al flux of $(8.6 \pm 2.5) \times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$ within the Inner Galaxy ($|\ell| \leq 30^{\circ}$, $|b| \leq 10^{\circ}$) with 3.7$σ$ significance above background. Within uncertainties, this flux is consistent with expectations from previous measurements by SPI and COMPTEL. This analysis demonstrates COSI's powerful capabilities for studies of $γ$-ray lines and underscores the scientific potential of future compact Compton telescopes. In particular, the next iteration of COSI as a NASA Small Explorer satellite has recently been approved for launch in 2025.