Heather Watson and Jo Konta both won in Tallin, and a doubles win took them to a 3-0 win.
Watson defeated Francisca Jorge 6-4 6-1.
Konta enjoyed a straights set victory of her own, defeating Maria Joao Koehler 6-1 6-0.
Fed-Cup Singles
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09/11/2018 AT 12:48
In the doubles, Koehler and Jorge teamed up against Anna Smith and Katie Boulter, with the Britons winning 6-0 6-2.
The next match for Britain is Estonia on Friday, and they would go on to play the winner of Group C in a play off should they finish top of their group.
If they won that tie then they would then take part in the World Group II play-offs, to be held in April.
Fed-Cup Singles
Venus gives U.S. lead against Netherlands
10/02/2018 AT 21:40
Fed-Cup Singles
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07/02/2018 AT 19:29