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We present the discovery of two small planets transiting HD 93963A (TOI-1797), a G0\,V star (M$_*$=1.109\,$\pm$\,0.043\,M$_\odot$, R$_*$=1.043\,$\pm$\,0.009\,R$_\odot$) in a visual binary system. We combined TESS and CHEOPS space-borne photometry with data from MuSCAT 2, `Alopeke, PHARO, TRES, FIES, and SOPHIE. We validated and spectroscopically confirmed the outer transiting planet HD 93963 Ac, a sub-Neptune with an orbital period of P$_c \approx$ 3.65 d, reported as a TESS object of interest (TOI) shortly after the release of Sector 22 data. HD 93963 Ac has a mass of M$_c = 19.2 \pm 4.1$ M$_{\oplus}$ and a radius of R$_c = 3.228 \pm 0.059$ R$_{\oplus}$, implying a mean density of $ρ_c=3.1\pm0.7$ gcm$^{-3}$. The inner object, HD 93963 Ab, is a validated 1.04 d ultra-short period (USP) transiting super-Earth that we discovered in the TESS light curve and that was not listed as a TOI, owing to the low significance of its signal (TESS signal-to-noise ratio $\approx$ 6.7, TESS $+$ CHEOPS combined transit depth D$_b=141.5 \pm 8.5$ ppm). We intensively monitored the star with CHEOPS by performing nine transit observations to confirm the presence of the inner planet and validate the system. HD 93963 Ab is the first small (R$_b = 1.35 \pm 0.042$ R$_{\oplus}$) USP planet discovered and validated by TESS and CHEOPS. Unlike planet c, HD 93963 Ab is not significantly detected in our radial velocities (M$_b = 7.8 \pm 3.2$ M$_{\oplus}$). We also discovered a linear trend in our Doppler measurements, suggesting the possible presence of a long-period outer planet. With a V-band magnitude of 9.2, HD 93963 A is among the brightest stars known to host a USP planet, making it one of the most favourable targets for precise mass measurement via Doppler spectroscopy and an important laboratory to test formation, evolution, and migration models of planetary systems hosting ultra-short period planets.