We investigate ambiguities in reconstructing a unitary elastic amplitude at fixed energy given the differential cross-section when one assumes analyticity in the cos ϕ plane. Strong hints are given that not more than a twofold nontrivial ambiguity is present. This is demonstrated for genuine entire functions of finite order. Moreover it is found that within a class of amplitudes which includes polynomials as well as entire functions of order zero, i) the difference between the two amplitudes with the same cross-section must be a polynomial, ii) if the cross-section is smaller than 1.38 (4π/k2) there is no ambiguity. Indications are given on directions for future work.