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In a functional calculus, the so called \Omega-rule states that if two termsP and Q applied to any closed term <i>N</i> return the same value (i.e. PN =QN), then they are equal (i.e. P = Q holds). As it is well known, in the\lambda\beta-calculus the \Omega-rule does not hold, even when the \eta-rule(weak extensionality) is added to the calculus. A long-standing problem of H.Barendregt (1975) concerns the determination of the logical power of the\Omega-rule when added to the \lambda\beta-calculus. In this paper we solve theproblem, by showing that the resulting theory is \Pi\_{1}^{1}-complete.

The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus