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We identify an equality between two objects arising from different contextsof mathematical physics: Kahane's Gaussian Multiplicative Chaos ($GMC^\gamma$)on the circle, and the Circular Beta Ensemble $(C\beta E)$ from Random MatrixTheory. This is obtained via an analysis of related random orthogonalpolynomials, making the approach spectral in nature. In order for the equalityto hold, the simple relationship between coupling constants is $\gamma =\sqrt{\frac{2}{\beta}}$, which we establish only when $\gamma \leq 1$ orequivalently $\beta \geq 2$. This corresponds to the sub-critical and criticalphases of the $GMC$.  As a side product, we answer positively a question raised by Virag. We alsogive an alternative proof of the Fyodorov-Bouchaud formula concerning the totalmass of the $GMC^\gamma$ on the circle. This conjecture was recently settled byR\'emy using Liouville conformal field theory. We can go even further anddescribe the law of all moments.  Furthermore, we notice that the ``spectral construction'' has a fewadvantages. For example, the Hausdorff dimension of the support is efficientlydescribed for all $\beta>0$, thanks to existing spectral theory. Remarkably,the critical parameter for $GMC^\gamma$ corresponds to $\beta=2$, where thegeometry and representation theory of unitary groups lie.

On the circle, Gaussian Multiplicative Chaos and Beta Ensembles match exactly