  We present new measurements of the diffuse radio halo emission from the Coma
cluster of galaxies at 2.675 GHz and 4.85 GHz using the Effelsberg 100--m
telescope. After correction for the contribution from point sources we derive
the integrated flux densities for the halo source (Coma C), $S_{2.675
GHz}=(107\pm 28)mJy$ and $S_{4.85 GHz}=(26\pm 12)mJy$. These values verify the
strong steepening of the radio spectrum of Coma C at high frequencies. Its
extent strongly depends on frequency, at 4.85 GHz it is only marginally
visible. The measurement at 4.85 GHz is the first flux density determination
for Coma C at this high frequency. In order to quantify the spectral steepening
we compare the expectations for the spectrum of Coma C with the observations,
resorting to basic models for radio halo formation. The in--situ acceleration
model provides the best fit to the data. From equipartition assumptions we
estimate a magnetic field strength $B_{eq}=0.57 (1+k)^{0.26}\mu G$ in the
intracluster medium of Coma, where k is the energy ratio of the positively and
negatively charged particles.
  As a by--product of the 2.675 GHz observation we present a new flux density
for the diffuse emission of the extended source 1253+275 ($S_{2.675GHz}=112\pm
10$). This measurement provides a smaller error range for the power--law fit to
the spectrum ($\alpha=1.18\pm 0.02$) compared to previous investigations and
yields an equipartition magnetic field strength of $B_{eq}=0.56 (1+k)^{0.24}\mu
G$.
