Scattered sunlight from the interplanetary dust (IPD) cloud in our solar system presents a serious foreground challenge for spectrophotometric measurements of the extragalactic background light (EBL). In this work, we report on inferred measurements of the absolute intensity of the zodiacal light (ZL) using the novel technique of Fraunhofer line spectroscopy on the deepest 8542 A line of the near-infrared Ca ii absorption triplet. The measurements are performed with the narrow band spectrometer (NBS) on board the Cosmic Infrared Background Experiment sounding rocket instrument. We use the NBS data to test the accuracy of two ZL models widely cited in the literature, the Kelsall and Wright models, which have been used in foreground removal analyses that produce high and low EBL results respectively. We find a mean reduced χ2 = 3.5 for the Kelsall model and χ2 = 2.0 for the Wright model. The best description of our data is provided by a simple modification to the Kelsall model, which includes a free ZL offset parameter. This adjusted model describes the data with a reduced χ2 = 1.5 and yields an inferred offset amplitude of 46 ± 19 nW m？2 sr？1 extrapolated to 12500 A. These measurements elude to the potential existence of a dust cloud component in the inner solar system whose intensity does not strongly modulate with the Earth's motion around the Sun.