We describe the NIRSpec component of the JWST Deep Extragalactic Survey (JADES), and provide deep spectroscopy of 253 sources targeted with the NIRSpec micro-shutter assembly in the Hubble Ultra Deep Field and surrounding GOODS-South. The multi-object spectra presented here are the deepest so far obtained with JWST, amounting to up to 28 hours in the low-dispersion ( R ~30–300) prism, and up to 7 hours in each of the three medium-resolution R ≈ 1000 gratings and one high-dispersion grating, G395H ( R ≈ 2700). Our low-dispersion and medium-dispersion spectra cover the wavelength range 0.6–5.3 μm. We describe the selection of the spectroscopic targets, the strategy for the allocation of targets to micro-shutters, and the design of the observations. We present the public release of the reduced 2D and 1D spectra, and a description of the reduction and calibration process. We measure spectroscopic redshifts for 178 of the objects targeted extending up to z = 13.2. We present a catalogue of all emission lines detected at S/N > 5, and our redshift determinations for the targets. Combined with the first JADES NIRCam data release, these public JADES spectroscopic and imaging datasets provide a new foundation for discoveries of the infrared universe by the worldwide scientific community.