American Airlines, British Airways, Iberia, Finnair and various other airlines have round-trip flights from Dallas (DFW) to London, United Kingdom (LHR) for $647-$683, NONSTOP or with a connection. Flights September to October. Departures on Monday, Tuesday, Wednesday, and Sunday. 7-day minimum stay required.
If both segments are Nonstop, the price is in the higher end of the range: $683-ish.
A Carry-on bag and also a personal item is allowed for free on American. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.