At Crypto'89, Chaum and van Antwerpen first introduced the concept of undeniable signatures, which has a special property such that a signature cannot be verified without the signer's cooperation. In 1996, Jakobsson, Sako, and Impagliazzo proposed a not-interactive undeniable signature scheme by employing a new primitive called designated verifier proofs. However, this paper shows that their scheme is insecure by demonstrating a simple attack that allows a dishonest signer to convince a designated verifier receiving invalid signatures. In addition, two intuitive countermeasures are presented.