Great Britain lost 2-1 to Hungary to finish third in their Fed Cup Europe/Africa Zone group in Budapest.
After a win for Heather Watson and a defeat for Johanna Konta, Jocelyn Rae and Tara Moore lost 5-7 7-5 6-3 to Timea Babos and Reka-Luca Jani in the doubles as the hosts secured victory.
Judy Murray's team had already seen their hopes of promotion to World Group II dashed by Friday's loss to Romania.
Britain were without number one Laura Robson because of a wrist injury.
They beat Latvia in their opening tie on Wednesday, but defeat in the key encounter against Romania meant Britain were playing for pride against Hungary.
Saturday's tie was still played in a tense atmosphere as they took on the hosts in front of a sizeable crowd that included Hungarian President Janos Ader.
Watson claimed her third singles victory of the week in the opening rubber as she reeled off the last five games in a row to beat Jani 6-4 6-1.
Konta then lost a dramatic second singles 4-6 7-6 (7-5) 6-4 to Babos after the Briton served for the match, and then led 5-1 in the second-set tie-break.
The British number two was tearful after 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, the British pair were pegged back and ultimately succumbed at 11.19pm local time.