Electronic cash (e-cash) refers to money exchanged electronically. The main features of traditional cash are usually considered desirable also in the context of e-cash. One such property is offline transferability, meaning the recipient of a coin in a transaction can transfer it in a later payment transaction to a third person without contacting a central authority. Among security properties, the anonymity of the payer in such transactions has been widely studied. This paper proposes the first efficient and secure transferable e-cash scheme with the strongest achievable anonymity properties, introduced by Canard and Gouget. In particular, it should not be possible for adversaries who receive a coin to decide whether they have owned that coin before. Our proposal is based on two recent cryptographic primitives: the proof system by Groth and Sahai, whose randomizability enables strong anonymity, and the commuting signatures by Fuchsbauer, which allow one to sign values that are only given as encryptions.