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Exoplanets can evolve significantly between birth and maturity, as their atmospheres, orbits, and structures are shaped by their environment. Young planets ($<$1 Gyr) offer an opportunity to probe the critical early stages of this evolution, where planets evolve the fastest. However, most of the known young planets orbit prohibitively faint stars. We present the discovery of two planets transiting HD 63433 (TOI 1726, TIC 130181866), a young Sun-like ($M_*=0.99\pm0.03$) star. Through kinematics, lithium abundance, and rotation, we confirm that HD 63433 is a member of the Ursa Major moving group ($τ=414\pm23$ Myr). Based on the TESS light curve and updated stellar parameters, we estimate the planet radii are $2.15\pm0.10R_\oplus$ and $2.67\pm0.12R_\oplus$, the orbital periods are 7.11 and 20.55 days, and the orbital eccentricities are lower than about 0.2. Using HARPS-N velocities, we measure the Rossiter-McLaughlin signal of the inner planet, demonstrating that the orbit is prograde. Since the host star is bright (V=6.9), both planets are amenable to transmission spectroscopy, radial velocity measurements of their masses, and more precise determination of the stellar obliquity. This system is therefore poised to play an important role in our understanding of planetary system evolution in the first billion years after formation.