Replacing old website with new one.

Hi! I am currently working on a website for somebody, with their current website sitting at domain.com. I want to upload the new one to domain.com and replace the old one. I had the new website temporarily sitting at domain.com/blog until it was done, and now it is. This is my first time doing this, and I would like to safely transfer everything without having much downtime for the company online.

Thank you!