  We present a catalogue of white dwarf candidates selected from the second
data release of $Gaia$ (DR2). We used a sample of spectroscopically confirmed
white dwarfs from the Sloan Digital Sky Survey (SDSS) to map the entire space
spanned by these objects in the $Gaia$ Hertzsprung-Russell diagram. We then
defined a set of cuts in absolute magnitude, colour, and a number of $Gaia$
quality flags to remove the majority of contaminating objects. Finally, we
adopt a method analogous to the one presented in our earlier SDSS photometric
catalogues to calculate a probability of being a white dwarf
($P_{\mathrm{WD}}$) for all $Gaia$ sources which passed the initial selection.
The final catalogue is composed of $486\,641$ stars with calculated
$P_{\mathrm{WD}}$ from which it is possible to select a sample of $\simeq
260\,000$ high-confidence white dwarf candidates in the magnitude range
$8<G<21$. By comparing this catalogue with a sample of SDSS white dwarf
candidates we estimate an upper limit in completeness of $85$ per cent for
white dwarfs with $G \leq 20$ mag and $T_{\mathrm{eff}} > 7000$ K, at high
Galactic latitudes ($|b|>20^{\circ}$). However, the completeness drops at low
Galactic latitudes, and the magnitude limit of the catalogue varies
significantly across the sky as a function of $Gaia$'s scanning law. We also
provide the list of objects within our sample with available SDSS spectroscopy.
We use this spectroscopic sample to characterise the observed structure of the
white dwarf distribution in the H-R diagram.
