This definition under the word (A2 Let verb (Rent)
http://dictionary.cambridge.org/dictionary/british/let_1 )
implies that in American English it might have the meaning you are thinking of.
In BrE we distinguish between the two meanings by using "rent" to mean you want to hire somewhere and pay money to do so, and the word "let" where you own somewhere and you wish to get money in from hiring out your property.
I want to rent a flat = I want to pay someone money to lease somewhere to live.
I want to let a flat = I own a flat, and I want to lease it out to someone who will pay me to live there. (in BrE)
But as I said, the dictionary shows that AmE uses "rent" in the way you are thinking.
One of our American English speaking members can let you know whether this is the case, and confirm how it works for them.