  The Murchison Widefield Array (MWA) has observed the entire southern sky
(Declination, $\delta <$ 30 deg) at low radio-frequencies, over the range
72-231 MHz. These observations constitute the GaLactic and Extragalactic
All-sky MWA (GLEAM) Survey, and we use the extragalactic catalogue (Galactic
latitude, $|b| >$ 10 deg) to define the GLEAM 4-Jy (G4Jy) Sample. This is a
complete sample of the 'brightest' radio-sources ($S_{\mathrm{151MHz}} >$ 4
Jy), the majority of which are active galactic nuclei with powerful radio-jets.
Crucially, low-frequency observations allow the selection of such sources in an
orientation-independent way (i.e. minimising the bias caused by Doppler
boosting, inherent in high-frequency surveys). We then use higher-resolution
radio images, and information at other wavelengths, to morphologically classify
the brightest components in GLEAM. We also conduct cross-checks against the
literature, and perform internal matching, in order to improve sample
completeness (which is estimated to be $>$ 95.5%). This results in a catalogue
of 1,863 sources, making the G4Jy Sample over 10 times larger than that of the
revised Third Cambridge Catalogue of Radio Sources (3CRR; $S_{\mathrm{178MHz}}
>$ 10.9 Jy). Of these G4Jy sources, 78 are resolved by the MWA (Phase-I)
synthesised beam ($\sim$2 arcmin at 200 MHz), and we label 67% of the sample as
'single', 26% as 'double', 4% as 'triple', and 3% as having 'complex'
morphology at $\sim$1 GHz (45-arcsec resolution). Alongside this, our
value-added catalogue provides mid-infrared source associations (subject to
6-arcsec resolution at 3.4 micron) for the radio emission, as identified
through visual inspection and thorough checks against the literature. As such,
the G4Jy Sample can be used as a reliable training set for cross-identification
via machine-learning algorithms. [Abstract abridged for arXiv submission.]
