Will verizon charge me for using Wi-Fi?

I just bought an iphone, and I have a 1GB a month data plan because I'm always in a Wi-Fi area. However, I just recently found out that if I do not turn off the cellular data (even if you keep the Wi-Fi connection on, ) you will still be charged for Data usage. Is this true? Will I have to keep my Celllular data off when I'm using Wi-Fi?

I have Verizon btw.

If youre feeding off of a Wi-Fi connection you shouldnt have to have your phones internet on, and you shouldnt be charged for data used from an independent Wi-Fi connection. At least i've never been charged for it.