At least 39 accounts were subject to the transactions and UBS was fined for systems and control failures that allowed the four members of the wealth management business to carry out the activity.
The incidents, which took place between January 2006 and December 2007, only came to light after a whistleblower raised concerns with UBS management.
An investigation by the FSA found that the employees had been allocating losses to customer accounts after trading foreign exchange and precious metals using their money.
UBS itself has estimated that around 50 such transactions a day were taking place.
Margaret Cole, FSA director of enforcement and financial crisis, said: "These employees were able to take advantage of UBS' inadequate systems and controls, giving them free rein to make unauthorized trades with customer money that they were then able to conceal."
Last month, the FSA fined UK investment bank Seymour Pierce $256,000 after one of its employees stole $249,000 from client accounts over a three-year period.
By Claire Archer