We detect Lyα absorption from the escaping atmosphere of HD 63433c, a R = 2.67R _⊕, P = 20.5 day mini-Neptune orbiting a young (440 Myr) solar analog in the Ursa Major Moving Group. Using Hubble Space Telescope (HST)/Space Telescope Imaging Spectrograph, we measure a transit depth of 11.1 ± 1.5% in the blue wing and 8 ± 3% in the red. This signal is unlikely to be due to stellar variability, but should be confirmed by an upcoming second transit observation with HST. We do not detect Lyα absorption from the inner planet, a smaller R = 2.15R _⊕ mini-Neptune on a 7.1 day orbit. We use Keck/NIRSPEC to place an upper limit of 0.5% on helium absorption for both planets. We measure the host star's X-ray spectrum and mid-ultraviolet flux with XMM-Newton, and model the outflow from both planets using a 3D hydrodynamic code. This model provides a reasonable match to the light curve in the blue wing of the Lyα line and the helium nondetection for planet c, although it does not explain the tentative red wing absorption or reproduce the excess absorption spectrum in detail. Its predictions of strong Lyα and helium absorption from b are ruled out by the observations. This model predicts a much shorter mass-loss timescale for planet b, suggesting that b and c are fundamentally different: while the latter still retains its hydrogen/helium envelope, the former has likely lost its primordial atmosphere.