  We argue that electron-electron interactions fundamentally restrict the
penetration length of Cooper pairs into a diffusive normal metal (N) from a
superconductor (S). At low temperatures this Cooper pair dephasing length
$L_\varphi$ remains finite and does not diverge at $T \to 0$. We evaluate the
subgap conductance of NS hybrids in the presence of electron-electron
interactions and demonstrate that this new length $L_\varphi$ can be directly
extracted from conductance measurements in such structures.
