The James Webb Space Telescope has discovered a surprising population of bright galaxies in the very early Universe (≲500 Myr after the Big Bang) that is hard to explain with conventional galaxy-formation models and whose physical properties are not fully understood. Insight into their internal physics is best captured through nebular lines, but at these early epochs, the brightest of these spectral features are redshifted into the mid-infrared and remain elusive. Using the mid-infrared instrument onboard the James Webb Space Telescope, here we present a detection of Hα and doubly ionized oxygen ([O iii] 4959,5007 Å) from the bright, ultra-high-redshift galaxy candidate GHZ2/GLASS-z12. Based on these emission lines, we infer a spectroscopic redshift of z = 12.33 ± 0.04, placing this galaxy just ~400 Myr after the Big Bang. These observations provide key insights into the conditions of this primaeval, luminous galaxy, which shows hard ionizing conditions rarely seen in the local Universe and probably driven by a compact and young burst (≲30 Myr) of star formation. The galaxy’s oxygen-to-hydrogen abundance is close to a tenth of the solar value, indicating a rapid metal enrichment. This study establishes the unique conditions of this notably bright and distant galaxy and the huge potential of mid-infrared observations to characterize these primordial systems.