We compute the exact value of the essential norm of ageneralized Hilbert matrix operator acting on weightedBergman spaces Apv and weighted Banach spaces H∞v ofanalytic functions, where v is a general radial weight. Inparticular, we obtain the exact value of the essential normof the classical Hilbert matrix operator on standard weightedBergman spaces Apα for p > 2 + α, α ≥ 0, and on Korenblumspaces H∞α for 0 < α < 1. We also cover the Hardy spaceHp, 1 < p < ∞, case. In the weighted Bergman space case, theessential norm of the Hilbert matrix is equal to the conjecturedvalue of its operator norm and similarly in the Hardy spacecase the essential norm and the operator norm coincide. Wealso compute the exact value of the norm of the Hilbert matrixon H∞wα with weights wα(z) = (1 − |z|)α for all 0 < α < 1. Also in this case, the values of the norm and essential normcoincide.

We relate the exponential integrability of the conjugate function (f) over tilde to the size of the gap in the essential range of f. Our main result complements a related theorem of Zygmund.