  Consider the Weil sum $W_{F,d}(u)=\sum_{x \in F} \psi(x^d+u x)$, where $F$ is
a finite field of characteristic $p$, $\psi$ is the canonical additive
character of $F$, $d$ is coprime to $|F^*|$, and $u \in F^*$. We say that
$W_{F,d}(u)$ is three-valued when it assumes precisely three distinct values as
$u$ runs through $F^*$: this is the minimum number of distinct values in the
nondegenerate case, and three-valued $W_{F,d}$ are rare and desirable. When
$W_{F,d}$ is three-valued, we give a lower bound on the $p$-adic valuation of
the values. This enables us to prove the characteristic $3$ case of a 1976
conjecture of Helleseth: when $p=3$ and $[F:{\mathbb F}_3]$ is a power of $2$,
we show that $W_{F,d}$ cannot be three-valued.
