We demonstrate generation of 7.6 fs near-UV pulses centered at 400 nm via 8-fold soliton-effect self-compression in an Ar-filled hollow-core kagomé-style photonic crystal fiber with ultrathin core walls. Analytical calculations of the effective compression length and soliton order permit adjustment of the experimental parameters, and numerical modeling of the nonlinear pulse dynamics in the fiber accurately predicts the spectrotemporal profiles of the self-compressed pulses. After compensation of phase distortion introduced by the optical elements along the beam path from the fiber to the diagnostics, 71% of the pulse energy was in the main temporal lobe, with peak powers in excess of 0.2 GW. The convenient setup opens up new opportunities for time-resolved studies in spectroscopy, chemistry, and materials science.