The business is part of the Coutts bank, owned by RBS, famous for counting Britain's royal family among its rich and famous customers.
The sale comes after RBS decided to split Coutts in two, keeping the UK-focused bank while hiving off a separate entity for business managed from Switzerland, Monaco, the Middle East, Singapore and Hong Kong.
Geneva-based UBP said it was acquiring a business with assets under management of more than 30 billion francs ($31 billion).
It did not disclose how much it was paying for the acquisition but the Financial Times has reported the bank, founded and owned by the de Picciotto family, was expected to pay $600 million to $800 million.
Coutts had been the Royal Bank of Scotland's international private banking and wealth management arm since 2000.
UPP said the acquisition was an "excellent strategic and geographic fit" as the international business of Coutts has a strong foothold in markets where, until this point, UBP had a limited presence.
It will enable the bank to "strengthen its presence in Switzerland and in Monaco, significantly boost its footprint in growth markets such as Central and Eastern Europe, the Middle East and Asia, and considerably increase its asset base in select European markets".
Guy de Picciotto, UBP’s CEO, said in a statement that the acquisition confirms the bank's commitment to further developing its wealth management business and represents a significant milestone in its growth strategy.
"This is particularly true for high-potential markets such as Asia, where the international business of Coutts has built long-standing relationships with high net worth clients," he said.
The transaction carried out with the support of Caurus Partners and TC Capital is subject to regulatory approvals.