  We present the characterization of KIC 4142768, an eclipsing binary with two
evolved A-type stars in an eccentric orbit with a period of 14 days. We measure
the fundamental parameters of the two components ($M_1=2.05M_{\odot},
R_1=2.96R_{\odot}$ and $M_2=2.05M_{\odot}, R_2=2.51R_{\odot}$) by combining
{\it Kepler} photometry and spectra from {\it Keck} HIRES. The measured surface
rotation rates are only one-fifth of the pseudo-synchronous rate of the
eccentric orbit. Fourier spectrum of the light curve reveals hybrid pulsations
of $\delta$ Scuti and $\gamma$ Doradus type, with pulsation frequencies at
about $15-18$ day$^{-1}$ for p modes and about $0.2-1.2$ day$^{-1}$ for
low-frequency g modes. Some of the g modes are exact orbital harmonics and are
likely tidally excited. Their pulsation amplitudes and phases both agree with
predictions from the linear tidal theory for $l=2, m=2$ prograde modes. We
examine the period spacing patterns in the free oscillating g modes and
identify them mostly as prograde sectoral dipole modes. The unstable frequency
range and frequency spacing of p modes and the inferred asymptotic g-mode
period spacings both agree with the stellar model for the primary star evolved
to a late stage of the main sequence. The inferred rotation rate of the
convective core boundary is very slow, similar to the small surface rotation
rate inferred from the spectroscopy. The measured surface and near-core
rotation rates provide constraints for testing the mechanism of angular
momentum transfer and tidal synchronization in evolved eccentric binary star
systems.
