The driver lost control and skidded off the track killing four women, one of whom was pregnant, and two men.
About a dozen people were hurt.
"One of the drivers took the track at a much higher speed than he should have and he veered off a curve, slamming into around 20 people," a Civil Guard spokesman said.
Television footage showed a rally car careering off a woodland track into a throng of onlookers in a cloud of dust.
Local media reported that the driver and co-driver were unharmed. The spokesman was unable to confirm that information.
"The Coruna Rally is in mourning. My deepest condolences to the bereaved and wishes for a complete recovery of those injured," Prime Minister Mariano Rajoy, who is from Galicia, said on his official Twitter feed.
(Reporting by Elisabeth O'Leary and Jesus Aguado; Editing by David Clarke)
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