Asymptotic giant branch stars are responsible for the production of most of the heavy isotopes beyond Sr observed in the solar system. Among them, isotopes shielded from the$r$-process contribution by their stable isobars are defined as$s$-only nuclei. For a long time the abundance of${}^{204}\mathrm{Pb}$, the heaviest$s$-only isotope, has been a topic of debate because state-of-the-art stellar models appeared to systematically underestimate its solar abundance. Besides the impact of uncertainties from stellar models and galactic chemical evolution simulations, this discrepancy was further obscured by rather divergent theoretical estimates for the neutron capture cross section of its radioactive precursor in the neutron-capture flow,${}^{204}\mathrm{Tl}$($t_{1/2}=3.78\mathrm{yr}$), and by the lack of experimental data on this reaction. We present the first ever neutron capture measurement on${}^{204}\mathrm{Tl}$, conducted at the CERN neutron time-of-flight facility n_TOF, employing a sample of only 9 mg of${}^{204}\mathrm{Tl}$produced at the Institute Laue Langevin high flux reactor. By complementing our new results with semiempirical calculations we obtained, at the$s$-process temperatures of$kT\approx 8\mathrm{keV}$and$kT\approx 30\mathrm{keV}$, Maxwellian-averaged cross sections (MACS) of 580(168) mb and 260(90) mb, respectively. These figures are about 3% lower and 20% higher than the corresponding values widely used in astrophysical calculations, which were based only on theoretical calculations. By using the new${}^{204}\mathrm{Tl}$MACS, the uncertainty arising from the${}^{204}\mathrm{Tl}(\mathrm{n},\gamma )$cross section on the$s$-process abundance of${}^{204}\mathrm{Pb}$has been reduced from$\sim 30\%$down to$+8\%/-6\%$, and the$s$-process calculations are in agreement with the latest solar system abundance of${}^{204}\mathrm{Pb}$reported by K. Lodders in 2021.