The European Union will continue to fund the Palestinian Authority in exchange for Hamas renouncing violence and recognising Israel.
A statement issued at the end of talks between EU foreign ministers in Brussels this evening said Europe "stands ready to continue to support Palestine's economic development and democratic state building".
But the statement also made clear that "at the same time", the terrorist group-turned democratic force in the Middle East was expected to drop its opposition to the existence of Israel, turn its back on violence and give up its arms.
The meeting was the first chance for EU ministers to decide a collective response to the surprise showing of Hamas in last Wednesday's poll - an election which Mr Straw said had unarguably been a democratic vote.
Some EU governments have been pressing for EU funding to be suspended in protest at the rise of a militant terrorist organisation to power. The EU is Palestine's biggest single donor, giving more than £340 million a year to the Palestinian Authority.
But, with the exact future role of Hamas in a new government still unclear, ministers avoided taking drastic action by cutting off financial aid and possibly damaging a delicate political balance in the Middle East.
The fact that senior Hamas officials urged the EU not to cut off funding was seen in Brussels as a positive sign that the organisation is willing to negotiate to reinforce its new democratic legitimacy.
But the rise of Hamas has created a major diplomatic dilemma for the EU - offering political recognition to a terror group still on an international blacklist.
Mr Straw made clear as he left to return to London that pressure is now on Hamas to respond to what is seen as a conciliatory but firm declaration from the EU.