We perform laser spectroscopy of a trapped ion that is not directly laser cooled. An ultracold atomic bath continuously cools the internal degrees of freedom of the ion to its hyperfine ground state via spin-exchange collisions while a co-trapped ${}^{174}$Yb${}^{+}$ ion sympathetically cools the external degrees of freedom. Successful laser excitation is detected via state-selective charge transfer following a collision with an atom and subsequent ion loss. As a demonstration of this spectroscopic technique, we probe the $6^2{S}_{1/2}\to 6^2{P}_{3/2}$ transition at 329~nm in ${}^{173}$Yb${}^{+}$ and measure the magnetic and electric hyperfine interaction constants for the $6^2{P}_{3/2}$ state to be $A=-241(1)$~MHz and $B=1460(8)$~MHz, respectively. Our results are in agreement with a previous measurement obtained in a hollow-cathode discharge experiment but are a factor of 6–9 more precise. The techniques demonstrated in this work may be extended to perform precision spectroscopy on ions with complex level structures with applications in fundamental physics.