We report the first measurement of a ratio λβb/λβc of bound-state (λβb) and continuum-state (λβc) β - decay rates for the case of bare Tl81+207 ions. These ions were produced at the GSI fragment separator FRS by projectile fragmentation of a Pb208 beam. After in-flight separation with the Bρ-ΔE-Bρ method, they were injected into the experimental storage-ring ESR at an energy of 400.5AMeV, stored, and electron cooled. The number of both the Tl81+207 ions and their bound-state β - decay daughters, hydrogenlike Pb81+207 ions, were measured as a function of storage time by recording their Schottky-noise intensities. The experimental result, λβb/λβc=0.188(18), is in very good agreement with the value of 0.171(1) obtained from theory employing spectra of allowed transitions.