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It is proved that there is a unique metrizable simplex $S$ whose extreme points are dense. This simplex is homogeneous in the sense that for every 2 affinely homeomorphic faces $F_1$ and $F_2$ there is an automorphism of $S$ which maps $F_1$ onto $F_2$. Every metrizable simplex is affinely homeomorphic to a face of $S$. The set of extreme points of $S$ is homeomorphic to the Hilbert space ${\ell }_2$. The matrices which represent $A\left(S\right)$ are characterized.