The £9m project at the National Library of Wales (NLW) in Aberystwyth was due to become home to 160,000 recordings from the BBC Wales programme archive.
Culture minister Lord Elis-Thomas has told the library he cannot commit £1m to the plans in their current form.
The move has been called "outrageous" by opposition politicians and NLW said it was "disappointed".
But the Welsh Government said the plan may put the library at long-term risk.
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