“Sprint Nextel Corp. will begin selling the iPhone 5 in mid-October, people familiar with the matter said, closing a huge hole in the No. 3 U.S. carrier’s lineup and giving Apple Inc. another channel for selling its popular phone.”

So begins the report in the Wall Street Journal — Apple’s (AAPL) current favorite backchannel outlet for releasing unofficial product news — that sparked a rally in Sprint’s (S) beleaguered shares. The stock soared $0.33 (10.12%) to close Tuesday at $3.59.

Apple also closed higher — up $17.16 (4.81%) to $373.60 — in a broad rally that pushed the Dow up nearly 3%.

According to the Journal, Sprint will sell both the forthcoming iPhone 5 and the existing iPhone 4 when the new device launches in mid-October.

The iPhone’s total available market in the U.S. will be nearly 300 million when Sprint’s 52 million subscribers are added to Verizon’s (VZ) 106 million and AT&T’s (T) 99 million.

Neither Sprint nor Apple could be reached for comment.