  We use lattice QCD to calculate the form factors $f_+(q^2)$ and $f_0(q^2)$
for the semileptonic decay $B_s\to K\ell\nu$. Our calculation uses six MILC
asqtad 2+1 flavor gauge-field ensembles with three lattice spacings. At the
smallest and largest lattice spacing the light-quark sea mass is set to 1/10
the strange-quark mass. At the intermediate lattice spacing, we use four values
for the light-quark sea mass ranging from 1/5 to 1/20 of the strange-quark
mass. We use the asqtad improved staggered action for the light valence quarks,
and the clover action with the Fermilab interpolation for the heavy valence
bottom quark. We use SU(2) hard-kaon heavy-meson rooted staggered chiral
perturbation theory to take the chiral-continuum limit. A functional $z$
expansion is used to extend the form factors to the full kinematic range. We
present predictions for the differential decay rate for both $B_s\to K\mu\nu$
and $B_s\to K\tau\nu$. We also present results for the forward-backward
asymmetry, the lepton polarization asymmetry, ratios of the scalar and vector
form factors for the decays $B_s\to K\ell\nu$ and $B_s\to D_s \ell\nu$. Our
results, together with future experimental measurements, can be used to
determine the magnitude of the Cabibbo-Kobayashi-Maskawa matrix element
$|V_{ub}|$.
