Fighting is reported to have spread to eastern Liberia after a mortar strike killed at least three people in a church filled with refugees in the capital Monrovia.
The mortar crashed through the roof, landing in the middle of the building and wounding many people.
The humanitarian situation grows worse by the day
The attack came despite a ceasefire declared by the rebels on Friday, shortly after the United States said it was sending a troop-carrying ship to the Liberian coast.
The BBC's Paul Welsh in Monrovia says there are reports of heavy fighting in the east of the country near Liberia's second biggest port, Buchanan.
The ceasefire between the government and the country's smaller rebel group Model in the east had been relatively stable before the reported artillery fire on Saturday.
In Monrovia, President Charles Taylor reiterated on Saturday that he would step down after international peacekeepers arrive - but did not say how soon after.
"Let nobody have any concern about, 'Will President Taylor step down?' I will step down," he told an independence day rally.
He has made such promises before.
The rally was called to mark the anniversary of the country's founding by freed American slaves in 1847.
He said either his vice president or the speaker of parliament would take over when he accepts an offer of refuge in Nigeria.
Rebels have continued firing indiscriminately into the centre of Monrovia, a day after eight people were killed when another mortar shell hit a school playground.
The United Nations has joined a chorus of approval for the US's decision to send warships to Liberia, but US President George Bush has said they will only support West African peacekeepers expected to arrive next week.
'US not ready'
UN Secretary General Kofi Annan wants the US to lead the force and to put a sizeable number of troops on the ground as soon as possible.
His spokesman Fred Eckhart said: "The US is not ready to make that commitment yet. But this commitment of logistical support is a big help and we'll just have to see if something else will follow."
The US is moving troops into position
The charity Oxfam said the US move would only be called a success "once peacekeepers are on the ground, people's lives are no longer threatened and aid operations are moving safely and freely".
Oxfam's Sam Nagbe in Monrovia added: "People are dying here every day. The Americans must play a leading role in an immediate peacekeeping intervention."
The BBC Pentagon correspondent Nick Childs says part of the problem for the Bush administration has been to devise some sort of limited mission that could still achieve clear goals.
More than a week of heavy fighting in Monrovia has left hundreds dead.
At least 25 people were killed in the city on Friday, including the eight who died when a mortar shell hit a school playground.
Monrovia has seen three onslaughts by rebels fighting to depose President Taylor in the past two months, resulting in desperate conditions for civilians.
Supplies of safe drinking water have run out in most parts of the city, increasing fears that cholera and other illnesses could spread rapidly.
Aid workers say the violence is preventing them reaching the thousands of Liberians crowded into churches, schools and other temporary shelters.