The neutron capture cross section of the unstable isotope $^{135}$Cs was measured relative to that of gold by means of the activation method. The sample was produced by ion implantation in a high resolution mass separator and irradiated with quasi-monoenergetic neutrons at 30 keV and 500 keV, using the $^{7}$Li(p,n)$^{7}$Be reaction. After the irradiations at the above energies, one more irradiation with thermal neutrons was used for defining the sample mass and for measuring the half-life of $^{136}$Cs. The neutron capture cross section was determined as 164 $\pm$ 10 mbarn and 34.8 $\pm$ 3.0 mbarn at 30 keV and 500 keV, respectively, and were used to normalize the theoretically derived cross section shape.