Judy Murray's team had already seen their hopes of promotion to World Group II dashed by defeat to Romania but they couldn't end on a positive note as Hungarians Timea Babos and Reka-Luca Jani won the doubles 5-7 7-5 6-3 to claim a 2-1 victory.
Watson got Great Britain off to a great stat against the hosts as she won the last five games in a row to beat Jani 6-4 6-1.
Konta then lost a dramatic second singles match 4-6 7-6 (7-5) 6-4 to Babos after the Briton had served for the match, and then led 5-1 in the second-set tie-break.
The British No.2 was left in tears at the defeat and, having also had treatment on her leg during the match, was replaced by Rae for the doubles.
However, after taking the first set, Rae and Moore were pegged back and slipped to defeat at 11.19pm local time.
- Sports & Recreation
- Heather Watson
- Johanna Konta
- Timea Babos
- Tara Moore
- Reka-Luca Jani