Let I be a regular proper ideal in a Noetherian ring R. We prove that there exists a simple free integral extension ring A of R such that the ideal I A has a Rees-good basis; that is, a basis c 1 , . . . , c g such that c i W = I W for i = 1, . . . , g and for all Rees valuation rings W of I A. Moreover, A may be constructed so that: (i) I A and I have the same Rees integers (with possibly different cardinalities), and (ii) A P is unramified over R P ∩R for each asymptotic prime divisor P of I A. Indeed, if H is a regular ideal in R such that each asymptotic prime divisor of H is contained in an asymptotic prime divisor of I, then (ii) holds for H A. If Card(Rees H) Card(Rees I), we prove that (i) also holds for H A and H. If I = (b 1 , . . . , b g )R and b 1 , . . . , b g is an asymptotic sequence, we prove that b 1 , . . . , b g is a Rees-good basis of I.