Fast radio bursts (FRBs) are millisecond-duration, luminous radio transients of extragalactic origin. These events have been used to trace the baryonic structure of the Universe using their dispersion measure (DM) assuming that the contribution from host galaxies can be reliably estimated. However, contributions from the immediate environment of an FRB may dominate the observed DM, thus making redshift estimates challenging without a robust host galaxy association. Furthermore, while at least one Galactic burst has been associated with a magnetar, other localized FRBs argue against magnetars as the sole progenitor model. Precise localization within the host galaxy can discriminate between progenitor models, a major goal of the field. Until now, localizations on this spatial scale have only been carried out in follow-up observations of repeating sources. Here we demonstrate the localization of FRB 20210603A with very long baseline interferometry (VLBI) on two baselines, using data collected only at the time of detection. We localize the burst to SDSS J004105.82+211331.9, an edge-on galaxy at , and detect recent star formation in the kiloparsec-scale vicinity of the burst. The edge-on inclination of the host galaxy allows for a unique comparison between the line of sight towards the FRB and lines of sight towards known Galactic pulsars. The DM, Faraday rotation measure (RM), and scattering suggest a progenitor coincident with the host galactic plane, strengthening the link between the environment of FRB 20210603A and the disk of its host galaxy. Single-pulse VLBI localizations of FRBs to within their host galaxies, following the one presented here, will further constrain the origins and host environments of one-off FRBs.