Although the production of X-rays from natural and rocket-triggered lightning leaders have been studied in detail over the last 10 years, the energy spectrum of the X-rays has never been well measured because the X-rays are emitted in very short but intense bursts that result in pulse pileup in the detectors. The energy spectrum is important because it provides information about the source mechanism for producing the energetic runaway electrons and about the electric fields that they traverse. We have recently developed and operated the first spectrometer for the energetic radiation from lightning. The instrument is part of the Atmospheric Radiation Imagery and Spectroscopy (ARIS) project and will be referred to as ARIS-S (ARIS Spectrometer). It consists of seven  NaI(Tl)/photomultiplier tube scintillation detectors with different thicknesses of attenuators, ranging from no attenuator to more than  of lead placed over the detector (all the detectors are in a  thick aluminum box). Using X-ray pulses preceding 48 return strokes in 8 rocket-triggered lightnings, we found that the spectrum of X-rays from leaders is too soft to be consistent with Relativistic Runaway Electron Avalanche. It has a power law dependence on the energies of the photons, and the power index, λ, is between 2.5 and 3.5. We present the details of the design of the instrument and the results of the analysis of the lightning data acquired during the summer of 2012.