The Nicole Farhi designer label is being sold to private equity group Open Gate Capital for up to £5m.
French Connection also plans to close 17 loss-making stores in the US.
French Connection chief executive Stephen Marks says: “tough decisions” are needed to get the business back on track.
The fashion retailer made a £16.4m loss in 2008.
The company says its restructuring programme is now complete and it is moving towards being “a more focused business which is expected to be profitable and cash generative, providing a solid base from which to develop the retained brands”.
In February, French Connection appointed Fallon to handle its advertising and launched a cross-platform ad campaign including TV and radio in February for its French Connection Man and Woman ranges.