  This Letter presents evidence for single top-quark production in the
$s$-channel using proton-proton collisions at a centre-of-mass energy of 8 TeV
with the ATLAS detector at the CERN Large Hadron Collider. The analysis is
performed on events containing one isolated electron or muon, large missing
transverse momentum and exactly two $b$-tagged jets in the final state. The
analysed data set corresponds to an integrated luminosity of 20.3 fb$^{-1}$.
The signal is extracted using a maximum-likelihood fit of a discriminant which
is based on the matrix element method and optimized in order to separate
single-top-quark $s$-channel events from the main background contributions,
which are top-quark pair production and $W$ boson production in association
with heavy-flavour jets. The measurement leads to an observed signal
significance of 3.2 standard deviations and a measured cross-section of
$\sigma_s=4.8\pm0.8$(stat.)$^{+1.6}_{-1.3}$(syst.) pb, which is consistent with
the Standard Model expectation. The expected significance for the analysis is
3.9 standard deviations.
