  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 ~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.
