  The {\em packing chromatic number} $\chi_{\rho}(G)$ of a graph $G$ is the
least integer $k$ for which there exists a mapping $f$ from $V(G)$ to
$\{1,2,\ldots ,k\}$ such that any two vertices of color $i$ are at distance at
least $i+1$. This paper studies the packing chromatic number of infinite
distance graphs $G(\mathbb{Z},D)$, i.e. graphs with the set $\mathbb{Z}$ of
integers as vertex set, with two distinct vertices $i,j\in \mathbb{Z}$ being
adjacent if and only if $|i-j|\in D$. We present lower and upper bounds for
$\chi_{\rho}(G(\mathbb{Z},D))$, showing that for finite $D$, the packing
chromatic number is finite. Our main result concerns distance graphs with
$D=\{1,t\}$ for which we prove some upper bounds on their packing chromatic
numbers, the smaller ones being for $t\geq 447$:
$\chi_{\rho}(G(\mathbb{Z},\{1,t\}))\leq 40$ if $t$ is odd and
$\chi_{\rho}(G(\mathbb{Z},\{1,t\}))\leq 81$ if $t$ is even.
