  We present a new calculation of the first Gegenbauer moment $a_1^K$ of the
kaon light-cone distribution amplitude. This moment is determined by the
difference between the average momenta of strange and nonstrange valence quarks
in the kaon. To calculate $a_1^K$, QCD sum rule for the diagonal correlation
function of local and nonlocal axial-vector currents is used. Contributions of
condensates up to dimension six are taken into account, including
$O(\alpha_s)$-corrections to the quark-condensate term. We obtain
$a_1^K=0.05\pm 0.02$, differing by the sign and magnitude from the recent
sum-rule estimate from the nondiagonal correlation function of pseudoscalar and
axial-vector currents. We argue that the nondiagonal sum rule is numerically
not reliable. Furthermore, an independent indication for a positive $a_1^K$ is
given, based on the matching of two different light-cone sum rules for the
$K\to\pi$ form factor. With the new interval of $a_1^K$ we update our previous
numerical predictions for SU(3)-violating effects in $B_{(s)}\to K$ form
factors and charmless (B) decays.
