  We present the first lattice QCD calculation of the form factor for B-> D* l
nu with three flavors of sea quarks. We use an improved staggered action for
the light valence and sea quarks (the MILC configurations), and the Fermilab
action for the heavy quarks. The form factor is computed at zero recoil using a
new double ratio method that yields the form factor more directly than the
previous Fermilab method. Other improvements over the previous calculation
include the use of much lighter light quark masses, and the use of lattice
(staggered) chiral perturbation theory in order to control the light quark
discretization errors and chiral extrapolation. We obtain for the form factor,
F_{B-> D*}(1)=0.921(13)(20), where the first error is statistical and the
second is the sum of all systematic errors in quadrature. Applying a 0.7%
electromagnetic correction and taking the latest PDG average for F_{B->
D*}(1)|V_cb| leads to |V_cb|=(38.7 +/- 0.9_exp +/- 1.0_theo) x 10^-3.
