This paper describes the third public data release (DR3) of the Calar Alto Legacy Integral Field Area (CALIFA) survey. Science-grade quality data for 667 galaxies are made public, including the 200 galaxies of the second public data release (DR2). Data were obtained with the integral-field spectrograph PMAS/PPak mounted on the 3.5 m telescope at the Calar Alto Observatory. Three different spectral setups are available: i) a low-resolution V500 setup covering the wavelength range 3745-7500 Å (4240-7140 Å unvignetted) with a spectral resolution of 6.0 Å (FWHM) for 646 galaxies, ii) a medium-resolution V1200 setup covering the wavelength range 3650-4840 Å (3650-4620 Å unvignetted) with a spectral resolution of 2.3 Å (FWHM) for 484 galaxies, and iii) the combination of the cubes from both setups (called COMBO) with a spectral resolution of 6.0 Å and a wavelength range between 3700-7500 Å (3700-7140 Å unvignetted) for 446 galaxies. The Main Sample, selected and observed according to the CALIFA survey strategy covers a redshift range between 0.005 and 0.03, spans the color-magnitude diagram and probes a wide range of stellar masses, ionization conditions, and morphological types. The Extension Sample covers several types of galaxies that are rare in the overall galaxy population and are therefore not numerous or absent in the CALIFA Main Sample. All the cubes in the data release were processed using the latest pipeline, which includes improved versions of the calibration frames and an even further improved image reconstruction quality. In total, the third data release contains 1576 datacubes, including ~1.5 million independent spectra. Based on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck-Institut für Astronomie (MPIA) and the Instituto de Astrofísica de Andalucía (CSIC).The spectra are available at http://califa.caha.es/DR3