  We performed a laser spectroscopic determination of the $2s$ hyperfine
splitting (HFS) of Li-like $^{209}\text{Bi}^{80+}$ and repeated the measurement
of the $1s$ HFS of H-like $^{209}\text{Bi}^{82+}$. Both ion species were
subsequently stored in the Experimental Storage Ring at the GSI
Helmholtzzentrum f\"ur Schwerionenforschung Darmstadt and cooled with an
electron cooler at a velocity of $\approx 0.71\,c$. Pulsed laser excitation of
the $M1$ hyperfine-transition was performed in anticollinear and collinear
geometry for $\text{Bi}^{82+}$ and $\text{Bi}^{80+}$, respectively, and
observed by fluorescence detection. We obtain $\Delta E^{(1s)}=
5086.3(11)\,\textrm{meV}$ for $\text{Bi}^{82+}$, different from the literature
value, and $\Delta E^{(2s)}= 797.50(18)\,\textrm{meV}$ for $\text{Bi}^{80+}$.
These values provide experimental evidence that a specific difference between
the two splitting energies can be used to test QED calculations in the
strongest static magnetic fields available in the laboratory independent of
nuclear structure effects. The experimental result is in excellent agreement
with the theoretical prediction and confirms the sum of the Dirac term and the
relativistic interelectronic-interaction correction at a level of 0.5%
confirming the importance of accounting for the Breit interaction.
