  We present the first search for the weak radiative decays $\Lambda_c^+ \to
\Sigma^+ \gamma$ and $\Xi_c^0 \to \Xi^0 \gamma$ using a data sample of
980~fb$^{-1}$ collected by the Belle detector operating at the KEKB
asymmetric-energy $e^+e^-$ collider. There are no evident $\Lambda_c^+ \to
\Sigma^+ \gamma$ or $\Xi_c^0 \to \Xi^0 \gamma$ signals. Taking the decays
$\Lambda_c^+ \to p K^- \pi^+$ and $\Xi_c^0 \to \Xi^- \pi^+$ as normalization
channels, the upper limits at 90\% credibility level on the ratios of branching
fractions ${\cal B}(\Lambda_c^+ \to \Sigma^+ \gamma)/{\cal B}(\Lambda_c^+ \to p
K^{-} \pi^+) < 4.0 \times 10^{-3}$ and ${\cal B}(\Xi_c^0 \to \Xi^0
\gamma)/{\cal B}(\Xi_c^0 \to \Xi^- \pi^+) < 1.2 \times 10^{-2}$ are determined.
We obtain the upper limits at 90\% credibility level on the absolute branching
fractions ${\cal B}(\Lambda_c^+ \to \Sigma^+ \gamma) < 2.6 \times 10^{-4}$ and
${\cal B}(\Xi_c^0 \to \Xi^0 \gamma) < 1.8 \times 10^{-4}$.
