  For a Banach space $X$, let $\mathcal{L}(X)$ denote the algebra of all
bounded linear operators on $X$ and let $\mathcal{K}(X)$ denote the compact
operator ideal in $\mathcal{L}(X)$. The quotient algebra
$\mathcal{L}(X)/\mathcal{K}(X)$ is called the Calkin algebra of $X$, and it is
denoted $\mathcal{C}al(X)$. We prove that the unitization of $\mathcal{K}(c_0)$
is isomorphic as a Banach algebra to the Calkin algebra of some Banach space
$\mathcal{Z}_{\mathcal{K}(c_0)}$. This Banach space is an Argyros-Haydon sum
$(\oplus_{n=1}^\infty X_n)_\mathrm{AH}$ of a sequence of copies $X_n$ of a
single Argyros-Haydon space $\mathfrak{X}_\mathrm{AH}$, and the external versus
the internal Argyros-Haydon construction parameters are chosen from disjoint
sets.
