  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 $\PP^r$. In
this note, we show that any non-empty $\mathcal{H}_{g+2,g,4}$ is irreducible
without any restriction on the genus $g$. Our result augments the
irreducibility result obtained earlier by Hristo Iliev(2006), in which several
low genus $g\le 10$ cases have been left untreated.
