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We present a uniqueness result in dimensions 3 for the inverse fixed angle scattering problem associated to the Schrödinger operator - Δ + q {-\Delta+q} , where q is a small real-valued potential with compact support in the Sobolev space W β , 2 {W^{\beta,2}} , with β > 0. {\beta>0.} This result improves the known result [P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations 17 1992, 55–68], in the sense that almost no regularity is required for the potential. The uniqueness result still holds in dimension 4, but for more regular potentials in W β , 2 {W^{\beta,2}} , with β > 2 / 3 {\beta>2/3}. The proof is a consequence of the reconstruction method presented in our previous work, [J. A. Barceló, C. Castro, T. Luque and M. C. Vilela, A new convergent algorithm to approximate potentials from fixed angle scattering data, SIAM J. Appl. Math. 78 2018, 2714–2736]. [ABSTRACT FROM AUTHOR]