  Let $C\subset \bold P^r$ be an integral projective curve. One defines the
speciality index $e(C)$ of $C$ as the maximal integer $t$ such that
$h^0(C,\omega_C(-t))>0$, where $\omega_C$ denotes the dualizing sheaf of $C$.
Extending a classical result of Halphen concerning the speciality of a space
curve, in the present paper we prove that if $C\subset \bold P^5$ is an
integral degree $d$ curve not contained in any surface of degree $< s$, in any
threefold of degree $<t$, and in any fourfold of degree $<u$, and if
$d>>s>>t>>u\geq 1$, then $ e(C)\leq
{\frac{d}{s}}+{\frac{s}{t}}+{\frac{t}{u}}+u-6. $ Moreover equality holds if and
only if $C$ is a complete intersection of hypersurfaces of degrees $u$,
${\frac{t}{u}}$, ${\frac{s}{t}}$ and ${\frac{d}{s}}$. We give also some partial
results in the general case $C\subset \bold P^r$, $r\geq 3$.
