Ab initio molecular dynamics simulations were performed with the aim to follow two scenarios for an excess electron in cold water clusters. In the first one, an electron is attached to a quenched neutral cluster. Such an electron, initially very delocalized and loosely bound, shrinks somewhat and increases its vertical detachment energy to 1-1.5 eV within several picoseconds. Unlike in warm liquid clusters, the electron in this cold system does not, however, reach a more compact and strongly bound structure. In contrast, if an equilibrated negatively charged water cluster with a well-localized excess electron is instantaneously quenched to .0 K, the electron remains strongly bound in a water cavity and practically does not change its size and binding energy. These results have important consequences for detailed interpretation of photoelectron spectroscopy measurements of electrons solvated in aqueous clusters and liquid water microjets.