  We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which
is the union of components whose general point corresponds to a smooth
irreducible and non-degenerate curve of degree $d$ and genus $g$ in
$\mathbb{P}^r$. In this article, we show that any non-empty
$\mathcal{H}_{g+1,g,4}$ has only one component whose general element is linear
normal unless $g=9$. If $g=9$, we show that $\mathcal{H}_{g+1,g,4}$ is
reducible with two components and a general element of each component is
linearly normal. This establishes the validity of a certain modified version of
an assertion of Severi regarding the irreducibility of $\mathcal{H}_{d,g,r}$
for the case $d=g+1$ and $r=4$.
