Ermotti, a 51-year-old from Switzerland's Italian-speaking Ticino region, was appointed interim CEO in September after Oswald Gruebel quit the bank over the trading losses.
Former Bundesbank president Weber will be proposed as chairman at the annual general meeting in May, a year earlier than planned.
Ermotti's relationship with Weber will be closely scrutinised as sources said the German had seen him as a fallback option and had explored other, outside candidates.
Weber said in a statement released through UBS
I know Sergio Ermotti personally. I had previously met him during my time at the Bundesbank. I have also spoken to him often in the last few weeks. I welcome his appointment as UBS's Group CEO and look forward to working together with him, he said.
The board originally said it might take up to six months to replace Gruebel, but pressure mounted for a decision ahead of an annual investor event on Thursday and as sources said top UBS managers jockeyed for position in the CEO race.
It was important for UBS after the resignation of Oswald Gruebel that there be no leadership vacuum, outgoing chairman Kaspar Villiger said.
UBS said the board backed the lender's strategy to scale back its investment bank and focus on its core wealth management business, which will be detailed at the investor event in New York.
A source familiar with the situation said Ermotti would not make more job cuts on Thursday after the bank announced recently 3,500 positions would go.
UBS shares were down 3 percent at 10.62 francs at 1:31 p.m. British time, compared with a 2.2 percent weaker European banking index <.SX7P>.
The news was also welcomed by analysts and investors.
Today's announcement is positive as it provides UBS with more certainty about its leadership and hence brings in stability, said Vontobel analyst Teresa Nielsen.
The appointment represents a rapid ascent for Ermotti, a former UniCredit deputy boss who joined UBS in April. Before UniCredit, he spent most of his early career at Merrill Lynch, where he rose to co-head of global equity markets.
Charming and well-dressed, Ermotti is seen as a bridge- builder whose style contrasts with that of the gruff Gruebel who upset Swiss politicians by lobbying against tough capital rules designed to prevent another state bailout of the bank.
It's a good step. It was making things difficult having an interim CEO with a chairman who was sort of on his way out as well, said Florian Esterer, senior portfolio manager at Swisscanto, which holds over 9 million UBS shares.
Ermotti's most urgent priority is to restore trust with UBS's wealthy private clients following the trading scandal and to scale back the investment bank amid a glut of new regulations for riskier activities.
No profit is worth more than our reputation as a bank, Sergio Ermotti told media in Zurich on Tuesday.
Ermotti has played up his Swiss roots, saying he is motivated by a desire to return his country's flagship bank to its former glory as it prepares to celebrate its 150th anniversary next year.
Near-fanatical about sports as a teen, Ermotti originally wanted to become a ski teacher, but decided against it after an apprenticeship at Corner Bank, a local Ticino lender.
Ermotti's first major task will be to present UBS's new strategy to investors on Thursday. He said he would give new targets and details on the bank's future risk appetite at the event.
Investment banking head Carsten Kengeter has so far been spared in the cull following the trading scandal that claimed Gruebel as well as equities co-heads Francois Gouws and Yassine Bouhara. Influential investor group Ethos has called for Kengeter to step down over the losses.
Ermotti told a news conference that speculation over Kengeter's future was inappropriate. Kengeter is part of the top UBS management team and would hold a presentation on the securities unit at Thursday's event, Ermotti said.
(Additional reporting by Emma Thomasson and Martin de Sa'Pinto; Editing by Mike Nesbit and Erica Billingham)