British Airways, Delta, KLM, and American have round-trip flights from Austin (AUS) to Basel or Gothenburg (BSL) for $574-$609 with a connection. Flights in April, August, September, October, and January.
Flight may operate as a BA/AA or Delta/KLM code-share.
There are a few cheap $538 Gothenburg flights departing on Oct 12 with returns on Oct 21 or 24.