  We show that for an irreducible cubic $f\in\mathbb Z[x]$ and a full norm form
$\mathbf N(x_1,\ldots,x_k)$ for a number field $K/\mathbb Q$ satisfying certain
hypotheses the variety $f(t)=\mathbf N(x_1,\ldots,x_k)\ne 0$ satisfies the
Hasse principle. Our proof uses sieve methods.
