Swansea Council says 800 applicants have applied for its apprenticeship scheme, of which twelve places are on offer.
It comes as a new report predicts 7,000 more people could become unemployed in Wales this year.
The survey by think tank the Institute for Public Policy Research also found that 41,000 extra under 25-year-olds in the UK will be out of work - a new record.
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Carwyn Jones announced at the Welsh Labour conference that he would step down in the autumn.
Leeanne Phillips went bald after suffering a miscarriage just six weeks into her pregnancy.
Labour says it will press ahead with plans to introduce national holidays for the patron saints of England, Wales, Scotland and Ireland.