Or in the contrapositive form
Is every one-dimensional UFD noetherian?
I know how to construct a non-noetherian UFD (polynomials in infinite number of variables over a field) and I know that it is even possible to construct a finite dimensional non-notherian UFD from here, but this example is 3-dimensional.
Since a noetherian one-dimensional UFD is a PID and vice versa, the question could be also rephrased as
Is every one-dimensional UFD a PID?
I guess the answer is negative, since it is a nice and elegant statement that have never heard about.