Several long-running radio programmes have also been dropped by the corporation from its Welsh language radio station, Radio Cymru.
This includes the daily news and current affairs programme Dros Ginio and the Saturday morning edition of the news programme Dros Frecwast.
One of the station’s longest running programmes, Ar y Marc will also be cut in the latest cost-saving measures. The football discussion programme has been on the air for 35 years.
Presenter John Hardy’s departure from BBC Radio Cymru will see Radio Cymru 2 fill the morning gap leading up to 07:00, when Dros Frecwast goes on air.
Radio Wales’ budgets will reduce by a similar amount to Radio Cymru, with the Sunday schedule being “simplified” following Roy Noble’s announcement he will be leaving the station in the spring.
Lynn Bowles’ Sunday programme will also end, and the volume of commissioned programmes will be reduced.
Public tours of BBC Central Square in Cardiff will stop in line with other BBC centres.
The National Union of Journalists (NUJ) called for UK government intervention over the cuts.
Laura Davison, NUJ general secretary, said the cuts were “devastating and demoralising”.
“The latest wave of cuts piles on the misery already being felt,” she said.
“BBC local radio attracts millions of listeners around the country each week.
“It is the BBC’s dedicated workforce that has made it a success, and the broadcaster will not be able to continue providing quality journalism and original programming without them.
“The more cuts the BBC makes, the more its services suffer.”
She added: “This is where the UK government must step in. The BBC has been undermined by years of underinvestment and interference.”