Isn't the Atachfalaya the deepest river in the world? I think the Mississipi runs into it because it's so deep.
The Atchafalaya is quite deep for an inland river, the channel is up to 90' in spots. When in flood, the Mississippi tends toward changing course into the Atchafalaya because it is a shorter (and, thus, steeper) route to the Gulf (and sea level).
But the river you are thinking of is probably the Saguenay, in Quebec. It flows into the St. Lawrence from the north, above Quebec. At this point, the St. Lawrence is a wide estuary and several hundred feet deep. But the Saguenay is a staggering 900' feet deep where it enters the St. Lawrence.
Actually, the Saguenay was gouged out by a glacier and is more accurately described as a fiord.