We present the first search for the weak radiative decays Λc+→ς+γ and Ξc0→Ξ0γ 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 Λc+→ς+γ or Ξc0→Ξ0γ signals. Taking the decays Λc+→pK-π+ and Ξc0→Ξ-π+ as normalization channels, the upper limits at 90% credibility level on the ratios of branching fractions B(Λc+→ς+γ)/B(Λc+→pK-π+)<4.0×10-3 and B(Ξc0→Ξ0γ)/B(Ξc0→Ξ-π+)<1.2×10-2 are determined. We obtain the upper limits at 90% credibility level on the absolute branching fractions B(Λc+→ς+γ)<2.6×10-4 and B(Ξc0→Ξ0γ)<1.8×10-4. © 2023 authors. Published by the American Physical Society.