By Gabriele Steinhauser And Matthew Dalton 

BRUSSELS--Greece struck a tenuous agreement for a four-month extension of its bailout Friday, removing immediate concerns over a potential exit from Europe's currency union but setting the stage for more tense negotiations over the country's financial future.

The deal, reached after weeks of often harsh exchanges between Athens and other European capitals, forced the left-wing government of Prime Minister Alexis Tsipras to temper many of the anti-austerity pledges that swept it into power last month.

The tough negotiations have frayed relationships between Greece and other governments, and particularly with Wolfgang Schäuble, the German finance minister.

"Now we hope that trust can grow again. There's a lot of room for growth, " Mr Schäuble said afterward.

Market rose on Friday as the agreement was reached, with the Stoxx Europe 600 edging up 0.6% to reach its highest level since November 2007. Both Germany's DAX index and the U.K.'s FTSE 100 climbed 0.4%. The euro was up against the dollar and the yen.

U.S. stocks also rose on the news, as the Dow industrials, the S&P 500 and the Russell 2000 small-company index hit all-time highs.

Greece's Finance Minister Yanis Varoufakis rejected claims that Friday's deal constituted a surrender of the government's anti-austerity promises. "Our pre-electoral program was about four years. This deal is about four months," he said.

The agreement to extend the bailout that was due to run out Feb. 28 should fend off immediate concerns about Greece's deteriorating finances and help stem an outflow of deposits from its banks.

However, concerns about Greece's ability to pay its debts will linger. The deal leaves a series of hurdles that could trigger further bouts of nervousness, with the first to be encountered next week.

Under the agreement, Greece has to present by Monday a list of budget cuts and economic overhauls, which has to pass the scrutiny of the supervisors of the bailout, the so-called troika of the European Commission, the European Central Bank and the International Monetary Fund. On Tuesday, finance ministers will review the proposals.

Only once these measures have been implemented will Athens received the next EUR7.2 billion slice of a EUR240 billion ($273 billion) bailout that has kept it afloat for almost five years.

Crucially, the four-month extension only takes Greece to the end of June, upping the pressure to negotiate a follow-up rescue deal in time for almost EUR7 billion in bond repayments to the ECB in July and August.

The Greek finance minister can take some small victories back to Athens. Greece will likely be allowed to run a smaller primary surplus--a measure of its budget balance that strips out interest payments--than the 3% of gross domestic product mandated by its bailout deal this year.

But the ministers' statement stressed that this concession had been made only because the economy has performed worse than expected.

"Today we have found partners among those who up until very recently looked at us with suspicion," said Mr. Varoufakis. "There are some partners who still look at us with suspicion."

Several ministers said that the economic deterioration was due to the uncertainty following Mr. Tsipras's election win last month.

The Athens government will also get to swap out some measures that had been set out in the existing bailout deal, but only if it can come up with substitutes that have the same financial and economic impact.

Most importantly, if the deal is concluded, it should allow Greek banks to once again tap the ECB for short-term funding--restoring a protection whose absence has been one reason for Greek depositors withdrawing their savings in recent weeks.

"As of today we're beginning to be co-authors of our destiny, co-authors of the reforms that we want to implement," Mr. Varoufakis said.

It isn't clear whether that will be enough to sell Friday's deal to a coalition in Athens that comprises leftist activists who have spent years campaigning against the country's bailout as well as right-wing nationalists focused on regaining sovereignty.

"Today's deal is a good first step, but largely theoretical. Can it get through Parliament?" said Mujtaba Rahman, Europe director political risk consultancy Eurasiagroup. "The governing coalition [in Athens] doesn't have a mandate to implement this deal."

After two-failed meetings over the past 10 days, in which ministers couldn't even agree on a joint statement, negotiations on Friday took a different tack.

Rather than immediately launching into a discussion among all 19 ministers, the talks were concentrated among a few key players, including Mr. Schäuble, French Finance Minister Michel Sapin, IMF Managing Director Christine Lagarde and ECB President Mario Draghi.

Mostly absent from those talks was Mr. Varoufakis, according to officials present. "He didn't play a role here," said one official, adding that at times the main players would congregate in the hallway between the French, IMF and ECB offices.

Only once the main players had agreed on a statement, including the four-month extension rather than the six months that Athens had initially requested, was the deal presented to the other ministers. After another round of discussions, they signed off on the statement.

After the meeting, Mr. Schäuble, the most ardent opponent of anti-bailout rhetoric coming from the government in Athens, complained about the political complications Greece's finances were causing other eurozone countries.

"Don't think that this is fun," Mr. Schäuble said.

Viktoria Dendrinou contributed to this article.

Write to Gabriele Steinhauser at gabriele.steinhauser@wsj.com, Viktoria Dendrinou at viktoria.dendinrou@wsj.com and Matthew Dalton at matthew.dalton@wsj.com

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