Scotts Cheap Flights has found flights from Austin to Dublin for as little as $496 round-trip. I’m not able to duplicate that low fare (my best is $648 on United + Aer Lingus), but searching more date pairs might turn it up…or it might already be gone.
Check your date pairs in Google Flights. These low fares are not available in July and August.
Aer Lingus, Delta, Air France, Lufthansa, Air Canada, and United are offering some low airfares from DFW (and other American cities) to Dublin, Ireland. A variety of date pairs (with varying airfares) have been found from March to June, and August through December.
This discovery from Scott’s Cheap Fares.
Carriers include KLM, Aer Lingus, Jetairfly, Norwegian Air, and WOW Air. There are some very inexpensive trans-Atlantic fares available right now.
Won’t Last! 10 Cities Round-Trip to Europe $407 to $614
WOW flights route through Iceland, which can be a nice stopover. Note that some of these airlines tack on extra fees for things like checked luggage, etc.