We study the magnetic-field-induced quantum phase transition from a gapped quantum phase that <?format ?>has no magnetic long-range order into a gapless phase in the spin-1/2 ladder compound <?format ?>bis(2,3-dimethylpyridinium) tetrabromocuprate (DIMPY). At temperatures below about 1 K, the specific heat in the gapless phase attains an asymptotic linear temperature dependence, characteristic of a Tomonaga-Luttinger liquid. Inelastic neutron scattering and the specific heat measurements in both phases are in good agreement with theoretical calculations, demonstrating that DIMPY is the first model material for an two-leg spin ladder in the strong-leg regime.