A question about ideal theory

Suppose R is a (commutative!) integral domain in which each finitely
generated prime ideal is
principal. Must then every finitely generated ideal be principal?

I remark that it is known that if every prime ideal is principal
[resp. finitely generated], then every
ideal is principal [resp. finitely generated]. But in order to adapt
the proofs to the present case,
it seems that one would need to look at ideals which are maximal with
respect to the property
of being finitely generated but not principal, and it is not clear
that such ideals exist.

.

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