How long does it take for the brown line on my belly to go away after having my baby?

Slowly. The brown line slowly decreases as your hormone levels changes back to a non pregnant state. However, if you breast feed the line will remain for a longer period of time.
Linea nigra. The linea nigra us produced due to melanocyte stimulating hormone made by the placenta (also causes darkening of nipples, etc...). It can take a few months after delivery for it to diasappear.