It's a little deeper than 3 - 4 feet, and not as deep as the 10 - 15 feet often cited. Here's a response to a post I did earlier that may prove enlightening:
Another Disneyland Urban Myth ripe for destruction:
Photo from MousePlanet
See those stairs from the bottom of the ROA to the dock? I count 10 steps. At 8" rise per step, that's 6.6 feet--to the top of the dock! Subtract about a foot (since the water doesn't go to the top of the dock), and your looking at a depth of about 5 1/2 feet. That's pretty consistent all the way 'round.
There's not reason, of course, for the River to be any deeper.