  The discrete Ces\`aro operator $ C $ acts continuously in various classical
Banach sequence spaces within $ \mathbb{C}^{\mathbb{N}}.$ For the
coordinatewise order, many such sequence spaces $ X $ are also complex Banach
lattices (eg. $c_0, \ell^p $ for $ 1 < p \leq \infty , $ and $ ces (p)$ for $ p
\in \{ 0 \} \cup ( 1, \infty )).$ In such Banach lattice sequence spaces, $ C $
is always a positive operator. Hence, its order spectrum is well defined within
the Banach algebra of all regular operators on $ X .$ The purpose of this note
is to show, for every $ X $ belonging to the above list of Banach lattice
sequence spaces, that the order spectrum $ \sigma_{\rm o} (C)$ of $ C $
coincides with its usual spectrum $ \sigma ( C)$ when $ C $ is considered as a
continuous linear operator on the Banach space $ X .$
