  We apply the diagram-technique formalism beyond the Hartree-Fock
approximation to a two-dimensional nearly ideal electron gas in a weak
perpendicular magnetic field. The case of an almost completely filled upper
Landau level (filling factor $\nu_0 \lesssim 1$) with a quantum number $N_0 \gg
1$ is considered. We uncover two regimes of renormalization by
electron-electron interactions. In the first regime, where $N^{1/2}_0(1-\nu_0)
\ll 1$, these interactions lead to a splitting of the Landau levels. In the
second regime, where $N^{1/2}_0(1-\nu_0) \gg 1$, apart of the splitting, a
renormalization of the bare Zeeman splitting occurs. The intermediate case
$N^{1/2}_0 (1-\nu_0)\approx 1$ cannot be studied within our approach. The
applicability of the Fermi-liquid description is investigated for both regimes.
