  Let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$
elements and $\mathbb{F}_{q^n}$ its extension of degree $n$. The conjecture of
Morgan and Mullen asserts the existence of primitive and completely normal
elements (PCN elements) for the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ for
any $q$ and $n$. It is known that the conjecture holds for $n \leq q$. In this
work we prove the conjecture for a larger range of exponents. In particular, we
give sharper bounds for the number of completely normal elements and use them
to prove asymptotic and effective existence results for $q\leq n\leq
O(q^\epsilon)$, where $\epsilon=2$ for the asymptotic results and
$\epsilon=1.25$ for the effective ones. For $n$ even we need to assume that
$q-1\nmid n$.
