  The {\em distinguishing number} of a group $G$ acting faithfully on a set $V$
is the least number of colors needed to color the elements of $V$ so that no
non-identity element of the group preserves the coloring. The {\em
distinguishing number} of a graph is the distinguishing number of its full
automorphism group acting on its vertex set. A connected graph $\Gamma$ is said
to have {\em connectivity 1} if there exists a vertex $\alpha \in V\Gamma$ such
that $\Gamma \setminus \{\alpha\}$ is not connected. For $\alpha \in V$, an
orbit of the point stabilizer $G_\alpha$ is called a {\em suborbit} of $G$.
  We prove that every connected primitive graph with infinite diameter and
countably many vertices has distinguishing number 2. Consequently, any
infinite, connected, primitive, locally finite graph is 2-distinguishable; so,
too, is any infinite primitive group with finite suborbits. We also show that
all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian
products of connected denumerable graphs of infinite diameter have
distinguishing number 2. All of our results follow directly from a versatile
lemma which we call The Distinct Spheres Lemma.
