The Extragalactic Background Light (EBL) captures the total integrated emission from stars and galaxies throughout the cosmic history. The amplitude of the near-infrared EBL from space absolute photometry observations has been controversial and depends strongly on the modeling and subtraction of the Zodiacal light foreground. We report the first measurement of the diffuse background spectrum at 0.8-1.7 um from the CIBER experiment. The observations were obtained with an absolute spectrometer over two flights in multiple sky fields to enable the subtraction of Zodiacal light, stars, terrestrial emission, and diffuse Galactic light. After subtracting foregrounds and accounting for systematic errors, we find the nominal EBL brightness, assuming the Kelsall Zodiacal light model, is 42.7+11.9/-10.6 nW/m2/sr at 1.4 um. We also analyzed the data using the Wright Zodiacal light model, which results in a worse statistical fit to the data and an unphysical EBL, falling below the known background light from galaxies at <1.3 um. Using a model-independent analysis based on the minimum EBL brightness, we find an EBL brightness of 28.7+5.1/-3.3 nW/m2/sr at 1.4 um. While the derived EBL amplitude strongly depends on the Zodiacal light model, we find that we cannot fit the spectral data to Zodiacal light, Galactic emission, and EBL from solely integrated galactic light from galaxy counts. The results require a new diffuse component, such as an additional foreground or an excess EBL with a redder spectrum than that of Zodiacal light.