We consider the configuration model of random graph consisting of N vertices. The degrees of vertices are drawn independently from power-law distribution with the exponent τ ∈ (1; 2) under the condition that the sum of vertex degree is equal to n. We show that if N; n → ∞ in such a way that n ζ( τ )N = O(N^1/τ), where ζ(x) is the Rimann‘s zeta-function, then the limit structure of the graph is the same typical structure as the random graph without the condition on the sum of degrees.