  This paper reports on a measurement of the double-polarization observable G
in $\pi^0$ photoproduction off the proton using the CBELSA/TAPS experiment at
the ELSA accelerator in Bonn. The observable G is determined from reactions of
linearly-polarized photons with longitudinally-polarized protons. The polarized
photons are produced by bremsstrahlung off a properly oriented diamond
radiator. A frozen spin butanol target provides the polarized protons. The data
cover the photon energy range from 617 to 1325 MeV and a wide angular range.
The experimental results for G are compared to predictions by the Bonn-Gatchina
(BnGa), J\"ulich-Bonn (J\"uBo), MAID and SAID partial wave analyses.
Implications of the new data for the pion photoproduction multipoles are
discussed.
