Must it be continuous? Note we are talking about the base distribution. The sampled distribution is discrete.
1) If the base distribution is continuous, drawing from it will get a new value (a new cluster mean, for example) every time.
2) But, you can nest Dirichlet processes. Which implies the base distribution can be discrete. (You can place a prior over your base distribution which is just another Dirichlet process).
Consider the Blackwell-Macqueen urn scheme.
H is a distribution over colors.
Start with an empty urn.
With probability alpha sample a color from H. Note, if H is continuous, you'll get a different color every time. If the H is discrete, you may draw colors already present in the urn.
With probability proportional to n-1, pick a ball from the urn. Put the ball back and another one of the same color.
edit: The paper above demonstrates this is possible. Any distribution drawn for the a DP with a discrete base distribution will have the same atoms as the base distribution, but with different weights.
Still, I wonder how the CRP would work under this scheme since there's a chance you'd select a non-empty table (analogously to the above urn scheme).