An action of a compact Lie group on a symplectic manifold M with symplectic form \(\omega\) is called Hamiltonian, if it preserves the form \(\omega\) and all vector fields which are generated by elements of the Lie algebra are Hamiltonian. Let \(G\times M\to M\) be such an action. Denote by MG the universal fibre space with fibre M. The author proves that there exists a closed 2-form on MG, the restriction of which to each fibre coincides with \(\omega\). If M is a Kähler manifold and the action is holomorphic, it may be achieved that the 2-form on MG has the type (1,1). For the computation of the equivariant cohomology \(H^*_ G(M)=H^*(MG;{\mathbb{C}})\) of a manifold M the Leray spectral sequence is used, the \(E_ 2\)-term of which has the form \(E_ 2^{p,q}=H^ p(BG)\otimes H^ q(M).\) The main theorem affirms that the spectral sequence is trivial, i.e. \(E_ 2^{p,q}=E_{\infty}^{p,q}.\) The proof is based on the connection between the cohomology groups of M and those of the fixed point set.