The deal should go through in the first quarter of 2010 after regulatory checks have been carried out.
As a result of the deal, Julius Baer's managed client assets will increase by around ten per cent, which it estimates will bring in an additional $155 million by August 2010.
Raymond Baer, Julius Baer's chairman, said: "Julius Baer is taking advantage of current market developments in acquiring a high quality, profitable asset with a strong track record."
ING Bank (Switzerland) will be rebranded and merged into Julius Baer's local operations in Switzerland and the business of its subsidiaries in Jersey and Monaco will be transferred into Julius Baer's existing operations.
The deal is the latest downsizing move by the ING group.
Last month it also sold its Australian and New Zealand life insurance and wealth management operations to ANZ for $1.6 billion.
By Tony Aynsley