The tie, which will almost certainly take place on clay, will be held on April 20-21 with the winners being promoted to the competition's second tier for the 2014 campaign.
Despite the long trip, it is not the worst draw GB could have received.
Captain Judy Murray had spoken of her desire for a home tie - something GB have not had for 20 years in the competition - but she will have been happy to have avoided Poland in Wednesday morning's draw; the Poles have world number four Agnieszka Radwanska in their ranks.
In contrast, Argentina's top player, Paula Ormaechea, is currently ranked 195th in the world and she is facing a fitness fight to play in April's tie after picking up an injury during the weekend's defeat to Sweden.
Argentina led that tie 2-0 but lost the last three rubbers.
Florencia Molinero, the world number 214, was their other key player in that tie.
GB's Heather Watson and Laura Robson are both ranked inside the world's top 50 but neither is most at home on a claycourt.
Captain Murray said: "Argentina away on clay will be a tough challenge for us but we have a strong squad and a huge opportunity to grab a place in the World Group. Vamos!"
Watson, Robson, Johanna Konta and Anne Keothavong led GB into the play-offs by coming through last week's Europe/Africa Zone Group I matches in Israel.
If fit, the quartet are expected to contest April's play-off although former British number one Elena Baltacha could come into contention if she has recovered from ankle surgery by then.
The team does not have to be named until April 10 with a decision on the venue due later this month.
World Group II play-offs:
Belgium vs Poland
France vs Kazakhstan
Argentina vs Great Britain
Ukraine vs Canada
World Group I play-offs:
Germany vs Serbia
Australia vs Switzerland
Spain vs Japan
USA vs Sweden