We consider the problem of the construction of a measure supported on a real line from prescribed moments. The main idea is to use the auxiliary dynamical system with the discrete time associated with a semi-infinite Jacobi matrix. Then the inverse dynamic data for this system, so called response operator (discrete analog of a dynamic Dirichet-to-Neumann map) is given in terms of moments, and we can use ideas of the Boundary Control method to recover the spectral data, i.e. the measure of a truncated moments problem, from dynamic one. The remarkable fact is that in our procedure we do not use the Jacobi matrix itself. We also formulate the results on the uniqueness of the solution of Hamburger and Stieltjes moments problems.