Air Canada and United have round-trip flights from Austin (AUS) to Reykjavik (KEF) for $536-$545 with a connection. 7-day minimum stay required. Flights in September. Departures on Tuesday and Wednesday.
Note: There are Priceline “Express Deals” available for about $429. The airline and exact flight is hidden until after purchase. Purchases are final. No 24-hour free cancellation. Guaranteed to be on a major airline. Select the deal with 0-to-1 stops and a layover less than 5-hours.