  A metric space $\mathrm{M}=(M;\de)$ is {\em homogeneous} if for every
isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of $\mathrm{M}$
there exists an isometry of $\mathrm{M}$ onto $\mathrm{M}$ extending $f$. A
metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if it is
homogeneous and separable and complete and if it isometrically embeds every
separable metric space $\mathrm{M}$ with $\dist(\mathrm{M})\subseteq
\dist(\boldsymbol{U})$. (With $\dist(\mathrm{M})$ being the set of distances
between points in $\mathrm{M}$.)
  The main results are: (1) A characterization of the sets
$\dist(\boldsymbol{U})$ for Urysohn metric spaces $\boldsymbol{U}$. (2) If $R$
is the distance set of an Urysohn metric space and $\mathrm{M}$ and
$\mathrm{N}$ are two metric spaces, of any cardinality with distances in $R$,
then they amalgamate disjointly to a metric space with distances in $R$. (3)
The completion of a homogeneous separable metric space $\mathrm{M}$ which
embeds isometrically every finite metric space $\mathrm{F}$ with
$\dist(\mathrm{F})\subseteq \dist(\mathrm{M})$ is homogeneous.
