We apply our technique of using a Rb-stabilized ring-cavity resonator to measure the frequencies of various spectral components in the 555.8-nm ${^{1}S}_{0}\ensuremath{\rightarrow}{^{3}P}_{1}$ line of Yb. We determine the isotope shifts with 60 kHz precision, which is an order-of-magnitude improvement over the best previous measurement on this line. There are two overlapping transitions, $^{171}\text{Y}\text{b}(1/2\ensuremath{\rightarrow}3/2)$ and $^{173}\text{Y}\text{b}(5/2\ensuremath{\rightarrow}3/2)$, which we resolve by applying a magnetic field. We thus obtain the hyperfine constants in the ${^{3}P}_{1}$ state of the odd isotopes with a significantly improved precision. Knowledge of isotope shifts and hyperfine structure should prove useful for high-precision calculations in Yb necessary to interpret ongoing experiments testing parity and time-reversal symmetry violation in the laws of physics.