He warned that Washington was on track to breach the current limit of $7.4 trillion in late September.
If the ceiling is not raised, "we can finance government operations no longer than mid-to-late November," he said.
The move is likely to reignite controversy over the US's spiralling budget deficit.
US government expenditure, swollen by the war in Iraq, exceeded revenues by a record $375bn in 2003, and the gap is forecast to rise again to $445bn this year.
Washington's coffers have also been depleted by a round of controversial tax cuts, seen by critics as a boon to the rich.
Election issue
The opposition Democrats are likely to point to the country's deteriorating fiscal position as evidence that President George W Bush is mishandling the economy in the run-up to the presidential election in November.
The debt ceiling has already been raised twice since President Bush took office in 2000.
However, analysts say Congress is likely to back Mr Snow's call for an increase in the limit, as failure to do so could force the government to default on its debt, triggering chaos on global financial markets.
The US government has never defaulted on its debt in its 228-year history.
The Treasury also has scope to raise extra cash by juggling its accounts.
The US national debt - the sum of previous annual budget deficits - currently stands at nearly $7.3 trillion (£3.9 trillion).
According to US Treasury figures, the government will have to borrow about $89 billion in the three months to September, and a further $122 billion in the final quarter of 2004.
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