We show that for arbitrary convex functions the order of approximation (in the metric $C[a,b]) by rational functions of degree no higher than $n $ does not exceed the quantity $ C\cdot M\cdot\frac{\ln^2n}n $ ($ C $ an absolute constant, $ M$ the maximum modulus of the convex function). We prove also the existence of a~convex function whose order of approximation is greater than $\frac1{n\ln^2n}$.