We present the first major release of data from the SAMI Galaxy Survey. This data release focuses on the emission-line physics of galaxies. Data Release One includes data for 772 galaxies, about 20% of the full survey. Galaxies included have the redshift range 0.004 < z < 0.092, a large mass range (7.6 < log(Mstellar/M$_odot$) < 11.6), and star-formation rates of 10^-4 to 10^1 M$_odot$/yr. For each galaxy, we include two spectral cubes and a set of spatially resolved 2D maps: single- and multi-component emission-line fits (with dust extinction corrections for strong lines), local dust extinction and star-formation rate. Calibration of the fibre throughputs, fluxes and differential-atmospheric-refraction has been improved over the Early Data Release. The data have average spatial resolution of 2.16 arcsec (FWHM) over the 15~arcsec diameter field of view and spectral (kinematic) resolution R=4263 (sigma=30km/s) around Halpha. The relative flux calibration is better than 5% and absolute flux calibration better than $pm0.22$~mag, with the latter estimate limited by galaxy photometry. The data are presented online through the Australian Astronomical Observatorys Data Central.
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