  We prove that the jacobian of a hyperelliptic curve $y^2=(x-t)h(x)$ has no
nontrivial endomorphisms over an algebraic closure of the ground field $K$ of
characteristic zero if $t \in K$ and the Galois group of the polynomial $h(x)$
over $K$ is "very big" and $deg(h)$ is an even number >8. (The case of odd
$deg(h)>3$ follows easily from previous results of the author.)
