After a slow start — and a flood of gray market devices — official sales gather momentum

iPhonAsia‘s Dan Butterfield, citing Chinese press accounts, reported early Tuesday that sales Apple’s AAPL iPhone in China have passed the 300,000 mark.

It took China Unicom CHU , Apple’s official Chinese carrier, 40 days to sell its first 100,000 iPhones and less than 20 to reach 300,000.

“The Unicom iPhone Express has steadily picked up speed and is now steaming down the tracks,” wrote Butterfield.

The acceleration in sales follows a high-profile advertising campaign and a 46-city iPhone roadshow and education campaign launched last week.

Apple and Unicom have not had an easy time getting a foothold in the world’s largest cellphone market (nearly 720 million subscribers). Not only are they competing against gray-market and counterfeit iPhones that may number in the millions, but the device they are selling is perceived as “crippled.” In order to comply with government restrictions on high-speed Internet devices, the Apple had to build a version of the iPhone for the Chinese market that doesn’t have a Wi-Fi chip.

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[Follow Philip Elmer-DeWitt on Twitter @philiped]