The organisation hopes the deal will lead to the logo becoming as widespread as the Fairtrade symbol is on coffee and chocolate.
The identity of the retailer will be announced in the next few weeks.
The BUAV is also in talks with overseas corporations such as the American supermarket chain Whole Foods in an effort to make the bunny logo a globally recognised standard.
"The cosmetics and household products industries have global partnerships, and it is time we did too," said Michelle Thew, chief executive of the BUAV.
Thew, who joined the organisation in November, hopes that, by choosing products that carry the bunny logo, consumers will exert the same kind of pressure on retailers as they have done through the Fairtrade movement.
Companies that sign up to the bunny standard agree to stop all animal tests on their products and to stop purchasing ingredients that have been tested on animals. They also agree to an independent audit.
UK corporations agreed to a voluntary ban on the testing of cosmetics on animals in 1998 but, according to the BUAV, they still sell products that have been tested on animals abroad. UK companies also continue to test household products such as washing powder on animals.