We demonstrate a technique for frequency measurements of UV transitions with sub-MHz precision. The frequency is measured using a ring-cavity resonator whose length is calibrated against a reference laser locked to the D₂ line of ⁸⁷Rb. We have used this to measure the 398.8 nm ¹S₀ ↔ ¹P₁ line of atomic Yb. We report isotope shifts of all the seven stable isotopes, including the rarest isotope ¹⁶⁸Yb. We have been able to resolve the overlapping ¹⁷³Yb (F = 3/2) and ¹⁷²Yb transitions for the first time. We also obtain high-precision measurements of the excited-state hyperfine structure in the odd isotopes, ¹⁷¹Yb and ¹⁷³Yb. The measurements resolve several discrepancies among earlier measurements.