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Bilbao are the fourth oldest Spanish club and among three never to be relegated, alongside Real Madrid and Barcelona.
It was their 20th Copa del Rey title and first since 2014.
Real will fancy their chances of winning the trophy for a 20th time when they face Osasuna in the final after they beat Athletic Bilbao 2-1 on aggregate on Tuesday.
Messi scored twice as Barcelona thrashed Athletic Bilbao 4-0 in the Copa del Rey final on Saturday to win the trophy for a record-extending 31st time with all the goals coming in the last half an hour.
Danish forward Martin Braithwaite grabbed the decisive goal with a scrappy diving header early in the extra period.
Sociedad had already won the first leg 2-1 and cemented their spot in the decider when striker Mikel Oyarzabal scored a 41st-minute penalty after a handball in the area by a Mirandes player.
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