The loss-making chain was put on the market last month by the Icelandic investment giant as part of a strategy to concentrate on its larger retail assets, such as House of Fraser.
Chief executive Gunnar Sigurdsson said: "MK One no longer fitted our portfolio and we are pleased to announce the sale. MK One has a strong customer base and three good brands: Mk One, Sophie and Chapter. I wish Hilco every success."
Baugur revealed it had received more than 20 expressions of interest in MK One.
Financial details of the deal were not disclosed.