Theodore Beza, Calvin's successor at Geneva, obtained the codex from the monastery of St Irenaeus at Lyon after the city was sacked in 1562, and presented it to the University of Cambridge in 1581. It carries the shelfmark MS Nn.2.41 and has been in Cambridge ever since. Where the book was before Lyon is unknown. A note in the manuscript records its use at a council held at Lyon in 1546, and that is the earliest fixed point anybody has for it.
The unusual part is what Beza said when he handed it over. He was cautious about the codex's readings and wrote so, which is not what a donor normally does, and it is part of why the manuscript's reputation stayed mixed for three centuries afterwards. The text it carries diverges from the other early witnesses often and sometimes substantially, most of all in Acts, and for a long time that divergence was read as corruption rather than as evidence. Modern textual criticism treats the codex instead as the principal representative of a distinct early tradition, which is close to the opposite conclusion drawn from the same pages.
The library's own work on the book has not stopped. A leaf lost from the codex was recovered in the 2010s by imaging the offset impression it had left on the page it once faced, which is a reading taken from a stain rather than from ink, and a reminder that a custodian's holding can still yield text nobody knew was there.