We study temperature chaos in a two-dimensional Ising spin glass with random quenched bimodal couplings, by an exact computation of the partition functions on large systems. We study two temperature correlators from the total free energy and from the domain wall free energy: in the second case we detect a chaotic behavior. We determine and discuss the chaos exponent and the fractal dimension of the domain walls.PACS numbers: 75.10. Nr, 05.50.+q, 75.40.Gb, 75.40.Mg Introduction -A characteristic feature of spin glasses is the presence of chaos under small changes in the quenched couplings, in the temperature, or in the magnetic field [1,2,3,4,5,6,7,8,9]. With the expression temperature chaos we refer to the fragility of the equilibrium states of a disordered system under small temperature changes. Let us consider two typical equilibrium configurations of such a system under the same realization of the quenched disorder: the first configuration is in equilibrium at temperature T , while the second one is in equilibrium at temperature T ′ = T + ∆T . One says that there is temperature chaos if for arbitrarily small (but non-zero) values of ∆T , the typical overlap of two configurations at T and T ′ goes to zero when the system size diverges. The spatial distance ℓ(T, ∆T ) over which such overlaps decay is called the chaos length, and, as we will discuss better in the following, it scales as ℓ ∼ ∆T