  The dynamics of the $\eta\to 3\pi^0$ decay have been studied with the Crystal
Ball multiphoton spectrometer and the TAPS calorimeter. Bremsstrahlung photons
produced by the 1.5-GeV electron beam of the Mainz microtron MAMI-C and tagged
by the Glasgow photon spectrometer were used for $\eta$-meson production. The
analysis of $3 \times 10^6$ $\gamma p \to \eta p \to 3\pi^0 p \to 6\gamma p$
events yields the value $\alpha=-0.032\pm0.003$ for the $\eta\to 3\pi^0$ slope
parameter, which agrees with the majority of recent experimental results and
has the smallest uncertainty. The $\pi^0\pi^0$ invariant-mass spectrum was
investigated for the occurrence of a cusplike structure in the vicinity of the
$\pi^+\pi^-$ threshold. The observed effect is small and does not affect our
measured value for the slope parameter.
