We study the bipartite entanglement entropy for one-dimensional systems. Its qualitative behavior is quite well understood: for gapped systems the entropy saturates to a finite value, while it diverges logarithmically as the logarithm of the correlation length as one approaches a critical, conformal point of phase transition. Using the example of two integrable models, we argue that close to non-conformal points the entropy shows a peculiar singular behavior, characteristic of an essential singularity. At these non-conformal points the model undergoes a discontinuous transition, with a level crossing in the ground state and a quadratic excitation spectrum. We propose the entropy as an efficient tool to determine the discontinuous or continuous nature of a phase transition also in more complicated models. - F. Franchini, A. R. Its, B.-Q. Jin, V. E. Korepin; J. Phys. A 40:8467 (2007) - F. Franchini, A. R. Its, V. E. Korepin; J. Phys. A: Math. Theor. 41:025302 (2008) - F. Franchini, E. Ercolessi, S. Evangelisti, F. Ravanini; Phys. Rev. B 83:012402 (2011)