  We prove that in odd characteristic the jacobian of a hyperelliptic curve
$y^2=f(x)$ has no nontrivial endomorphisms over an algebraic closure of the
ground field if the Galois group of the polynomial $f$ of even degree is ``very
big". The case of characteristic zero was previously treated by the author
(Math. Res. Letters 7(2000), 123--132).
