Britain had led the best of three tie after Heather Watson secured an impressive (7)6-7 6-1 6-4 win over Soran Cirstea in Budapest.
But Simona Halep downed Johanna Konta 6-1 6-4 to level the tie before Romanian pair Irina-Camelia Begu and Monica Niculescu beat Watson and Jocelyn Rae in the doubles to take a commanding lead in Pool B.
The winner of each of four pools plays another to determine to two teams reach the World Group II play-offs in April.
Britain still have a chance of reaching the next phase, but they must beat hosts Hungary, hope Romania lose to Latvia - who GB have beaten - and sweat on the game average going their way.
Judy Murray's British squad were up against it the moment their top player Laura Robson was ruled out with a wrist injury, with hopes resting on the shoulders of Watson, who has struggled for form and fitness since a bout of glandular fever last year.
Watson, who has slipped down to 162nd in the world rankings caused an upset when beating world No. 26 Cirstea as she came from a set down to give GB a 1-0 lead.
The 21-year-old had her fair share of chances to claim the first stanza, but missed four set points – the first when serving at 6-5 and the other three during the tie-break.
However, despite losing the set Watson soon found her range and raced through the second – allowing Cirstea just the solitary game.
There was some controversy in the third and final set as Watson served a double fault at 3-3, but the ball burst as she made contact and the point was ordered to be replayed.
The Brit went on to win the game and soon claimed the match.
"It was an exciting match," Watson said. "She played great on the set points but I took charge of rallies after and that's what I'm most pleased with."
Watson, who reached a career-high of world no.39 a year ago before illness, has beaten high-ranked players before; the task was always going to be tougher for Konta, ranked 118th, against world number 10 Halep.