The fourth moment of Riemann’s zeta-function on the critical line can be studied in terms of integrals of the form \[ \int_{-\infty }^{\infty }| \zeta (1/2+it)| ^4g(t)\, dt, \] where \(g(t)\) is a “nice” function. A spectral decomposition for this expression has been given by the second named author [“Spectral theory of the Riemann zeta-function”, Cambridge Tracts in Math. 127 (1997; Zbl 0878.11001)]; previously he had derived an approximate formula of the same flavour [Acta Math. 170, 181–220 (1993; Zbl 0784.11042)]. The spectral decomposition arises via Kloosterman sums and the Kloosterman-Spectral sum formula due to N. V. Kuznetsov. The Kloosterman sums play an auxiliary role in the argument: first they appear and then disappear. The main motivation of the present paper is that such a detour can be avoided and the spectral decomposition can be derived directly. Instead of applying the spectral theory of Kloosterman sums, the authors make use of the spectral structure of the space \(L^2(\Gamma \backslash G)\) with \(\Gamma = \text{PSL}_2(\mathbb Z)\) and \(G= \text{PSL}_2(\mathbb R)\). The main tool is a certain Poincar\acuteaccent e series over \(G\) whose value at the unit element is related to the non-diagonal part of the fourth moment in question.