We present strong constraints on the spacetime variation of the fine-structure constant $\alpha$ using the Dark Energy Spectroscopic Instrument (DESI). In this pilot work, we utilize $\sim110,000$ galaxies with strong and narrow O III $\lambda\lambda$4959,5007 emission lines to measure the relative variation $\Delta\alpha/\alpha$ in space and time. The O III doublet is arguably the best choice for this purpose owing to its wide wavelength separation between the two lines and its strong emission in many galaxies. Our galaxy sample spans a redshift range of $0Comment: 17 pages, 11 figures, accepted for publication in ApJ