Drying up Breastmilk With contaceptive pill?

i have chosen to give up breastfeeding, my bub already goes 10 hours without a feed as she sleeps through the night and over the past few days iv cut down to 2-3 breastfeeds a day, i can already tell my breasts arent getting as full, my doctor put me on the combined contaceptive pill to help dry it up, i started taking them yesterday, if i only give one feed today then stop completely how long should it take for my milk supply to stop, considering iv started taking the pill?