We started a program to search for companions around hybrid \delta Sct/ \gamma Dor stars with the frequency modulation method using Kepler data . Our best candidate was KIC 5709664 , where we could identify Fourier peaks with sidelobes , suggesting binary orbital motion . We determined the orbital parameters with the phase modulation method and took spectroscopic measurements to confirm unambiguously the binary nature with radial velocities . The spectra show that the object is a double-lined spectroscopic binary , and we determined the orbital solutions from the radial velocity curve fit . The parameters from the phase modulation method and the radial velocity fits are in good agreement . We combined a radial velocity and phase modulation approach to determine the orbital parameters as accurately as possible . We determined that the pulsator is a hybrid \delta Sct/ \gamma Dor star in an eccentric binary system with an orbital period of \sim 95 d and an eccentricity of 0.55 . The measured mass ratio is 0.67 . We analysed the pulsation content and extracted 38 frequencies with amplitudes greater than 20 \mu mag . At low frequencies , we found broad power excesses which are likely attributed to spots on the rotating surface of the lower-mass component . We inferred rotation periods of 0.56 and 2.53 d for the primary and secondary , respectively .