  Let $\ell \ne 3$ be a prime. We show that there are only finitely many cyclic
number fields $F$ of degree $\ell$ for which the unit equation $$\lambda + \mu
= 1, \qquad \lambda,~\mu \in \mathcal{O}_F^\times$$ has solutions. Our result
is effective. For example, we deduce that the only cyclic quintic number field
for which the unit equation has solutions is $\mathbb{Q}(\zeta_{11})^+$.
