Let Ω be a convex planar domain, with no curvature or regularity assumption on the boundary. Let N_θ(R) = card{RΩ_θ∩ℤ²}, where Ω_θ denotes the rotation of Ω by θ. It is proved that, up to a small logarithmic transgression, N_θ(R) = |Ω\R² + O(R^2/3), for almost every rotation. A refined result based on the fractal structure of the image of the boundary of Ω under the Gauss map is also obtained.