  Using staggered fermions and twisted boundary conditions, we calculate the K
meson semileptonic decay vector form factor at zero momentum transfer. The HISQ
formulation is used for the valence quarks, while the sea quarks are simulated
with the asqtad action (MILC N_f=2+1 configurations). For the chiral and
continuum extrapolation we use two-loop continuum CHPT, supplemented by
partially quenched staggered CHPT at one loop. Our result is f_+^{K\pi}(0) =
0.9667+-0.0023+-0.0033, where the first error is statistical and the second is
the sum in quadrature of the systematic uncertainties. This result is the first
N_f=2+1 calculation with two lattice spacings and a controlled continuum
extrapolation. It is also the most precise result to date for the vector form
factor and, although the central value is larger than previous unquenched
lattice calculations, it is compatible with them within errors. Combining our
value for f_+^{K\pi}(0) with the latest experimental measurements of K
semileptonic decays, we obtain |V_{us}| = 0.2238+-0.0009+-0.0005, where the
first error is from f_+^{K\pi}(0) and the second one is experimental. As a
byproduct of our calculation, we obtain the combination of low-energy constants
[C_{12}^r+C_{34}^r-(L_5^r)^2](M_\rho) = (3.62+-1.00)x10^{-6}.
