Completions of valuation rings

Let k be a field of characteristic zero, K an algebraic function field over k, and V a k-valuation ring of K. Zariski's theorem of local uniformization shows that there exist algebraic regular local rings R_i with quotient field K which are dominated by V, and such that the direct union of the R_i's is V. We investigate the ring T, which is the direct union of the completions of the R_i's. We give necessary and sufficient conditions for T to be a valuation ring. We then focus on the case in which the valuation has rank one.