Apple began taking orders for the phone at midnight Pacific time, promising delivery by next Friday, when the new phone also goes on sale in stores.
Four hours later, the expected delivery time had grown to two weeks, according to Topeka Capital Markets analyst Brian White.
Less than four hours after that, Apple's website stopped taking orders, saying "We'll be back."
White says the quick rise in the expected delivery time suggests much stronger demand than Apple expected. Last year, one-week delivery of the iPhone 4S was available into the afternoon of the first order day.
"Clearly, iPhone 5 fever is in full swing," White says.
Apple's stock is up 1.7 percent to $694.29 in morning trading.