  The beam asymmetry, $\Sigma$, was measured at ELSA in the reaction $\vec
\gamma p \to \eta p$ using linearly polarised tagged photon beams, produced by
coherent bremsstrahlung off a diamond. The crystal was oriented to provide
polarised photons in the energy range $E_\gamma = 800$ to 1400 MeV with the
maximum polarisation of $P_\gamma = 49$ % obtained at 1305 MeV. Both dominant
decay modes of the $\eta$ into two photons and $3\pi^0$ were used to extract
the beam asymmetry from the azimuthal modulation of the cross section. The
measurements cover the angular range $\Theta_\text{cm}\simeq 50$ -- 150
degrees. Large asymmetries up to 80 % are observed, in agreement with a
previous measurement. The eta-MAID model and the Bonn--Gatchina partial wave
analysis describe the measurements, but the required partial waves differ
significantly.
