  The distinguishing number of a structure is the smallest size of a partition
of its elements so that only the trivial automorphism of the structure
preserves each cell of the partition. We show that for any countable subset of
the positive real numbers, the corresponding countable homogeneous Urysohn
metric space, when it exists, has distinguishing number 2 or the distinguishing
number is infinite.
  While it is known that a sufficiently large finite primitive structure has
distinguishing number 2, unless its automorphism group is the full symmetric
group or alternating group, the infinite case is open and these countable
Urysohn metric spaces provide further confirmation toward the conjecture that
all primitive homogeneous countably infinite structures have distinguishing
number 2 or else the distinguishing number is infinite.
