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We present the characterization of KIC 4142768, an eclipsing binary with twoevolved A-type stars in an eccentric orbit with a period of 14 days. We measurethe 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 surfacerotation rates are only one-fifth of the pseudo-synchronous rate of theeccentric orbit. Fourier spectrum of the light curve reveals hybrid pulsationsof $\delta$ Scuti and $\gamma$ Doradus type, with pulsation frequencies atabout $15-18$ day$^{-1}$ for p modes and about $0.2-1.2$ day$^{-1}$ forlow-frequency g modes. Some of the g modes are exact orbital harmonics and arelikely tidally excited. Their pulsation amplitudes and phases both agree withpredictions from the linear tidal theory for $l=2, m=2$ prograde modes. Weexamine the period spacing patterns in the free oscillating g modes andidentify them mostly as prograde sectoral dipole modes. The unstable frequencyrange and frequency spacing of p modes and the inferred asymptotic g-modeperiod spacings both agree with the stellar model for the primary star evolvedto a late stage of the main sequence. The inferred rotation rate of theconvective core boundary is very slow, similar to the small surface rotationrate inferred from the spectroscopy. The measured surface and near-corerotation rates provide constraints for testing the mechanism of angularmomentum transfer and tidal synchronization in evolved eccentric binary starsystems.

KIC 4142768: An Evolved Gamma Doradus/Delta Scuti Hybrid Pulsating Eclipsing Binary with Tidally Excited Oscillations