Oneworld Alliance airlines have Chicago to Stockholm or Gothenburg on sale for low $500s in July 2017. Iberia and Finnair offer routes with low fares when booked through online travel agency sites found through Momondo.com or Skyscanner.com. Some of these fares price in mid-$600s on Finnair site.
ExploreTrip.com ticket for Finnair flights
Chicago ORD – Helsinki HEL – Stockholm Bromma BMA – HEL – ORD
ExploreTrip.com ticket for Iberia flights
Chicago ORD – Madrid MAD – Stockholm ARN – Madrid MAD (17 hr. overnight layover) – Chicago ORD
Flygbussarna.com provides bus transportation between Stockholm’s airports Arlanda ARN and Bromma BMA to city at discount prices compared to train travel. Cheap travel on Norwegian, Ryanair and other low cost carriers are available from Stockholm airports to the rest of Europe.
Traveling to Europe in July via Stockholm can be far cheaper than flying directly to many European destinations when prices are typically over $1,000 round trip.