The sandwich chain has 500 UK outlets but has plans to extend this to 2,000 in the next five years, making it bigger than McDonald's in terms of store count.
Frame will create five campaigns, including a TV spot, over the next year. It aims to raise the profile of the brand as it expands in the UK.
The marketing spend behind the brand is expected to increase.
Deidre Anderson, the chair of Subway's national board, appointed Frame, which already works with the brand in the north of England.
Subway is in the process of recruiting its first UK marketing director.
An appointment is expected by early October.
In July, it appointed MediaCom's Scottish arm to handle its planning and buying.
Separately, David Miller, the former managing director of Wieden & Kennedy, has joined Frame as a partner.
He will work alongside the agency's two founders, Alan Frame and Graeme Atha.