We present James Webb Space Telescope (JWST) and Hubble Space Telescope (HST) observations of the counterpart of EP 240801a, at z = 1.67, the first fast X-ray transient (FXT) identified as an X-ray flash by the Einstein Probe and Fermi Gamma-Ray Burst Monitor. Our observations reveal strong photometric and spectroscopic evidence for an associated broad-lined Type Ic supernova (SN) SN 2024aihh, the most distant spectroscopically identified gamma-ray-burst (GRB)–SN to date. The SN exhibits similar luminosity and light-curve evolution to the prototype GRB-SN 1998bw with an absolute magnitude of the SN at ∼23 days rest frame of M _F140W ≈ −19 mag. The SN is located in a host galaxy with complex morphology at a large (∼6 kpc) offset in a region of relatively low surface brightness. The region around the SN has a modest star formation rate and is dominated by an intermediate mass-weighted age (1.4 ± 0.3 Gyr) population, despite the apparent presence of a young, massive broad-lined Type Ic SN progenitor. These observations demonstrate that observations with HST and JWST can greatly extend the redshift range over which the GRB/FXT-SN connection can be studied, including in relatively low-luminosity, X-ray-rich events. They demonstrate little apparent evolution in the SN properties from local examples despite EP 240801a originating from an epoch 10 Gyr ago.