The 2012 grand slam champions, who lost their opening match to Ireland, had needed to defeat previously unbeaten England by at least eight points, or seven as long as the visitors did not outscore them by three tries.
They went on to the attack from the opening whistle, dominating a frenetic first half, and taking a 9-3 lead at the interval through three Leigh Halfpenny penalties before cutting loose after the interval.
Cuthbert scored twice within 10 minutes to put the game beyond England's reach. The score was a record for Wales over England, breaking the previous best of 25-0 set in 1905.
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