Bru J.-B., Zagrebnov V.A. Exact Solution of the Bogoluibov Hamiltonian for Weakly Imperfect Bose-Gas (178K, LATeX) ABSTRACT. We show that the pressure for the Bogoluibov Weakly Imperfect Bose-Gas can be calculated exactly in the thermodynamic limit.We point out sufficient and necessary conditions that it does not coincide with the pressure of the Ideal Bose-Gas