  A primitive completely normal element for an extension
$\mathbb{F}_{q^n}/\mathbb{F}_{q}$ of Galois fields is a generator of the
multiplicative group of $\mathbb{F}_{q^n}$, which simultaneously is normal over
every intermediate field of that extension. We are going to prove that such a
generator exists when $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ is an 'exceptional'
regular extension. In combination with [6] our investigations altogether settle
the existence of primitive completely normal bases for any regular extension.
An important feature of the class of regular extensions is that they comprise
every extension of prime power degree.
