How are mini drill bits mounted on a regular sized drill?
One option that you have is to use a mini-drill chuck adapter or mini-drill chuck. I used the latter term in a search and the auto-fill in The Google added adapter, but the results were the same:



This image came from The Home Depot and will accept from 0.00 inches to 0.039 inches (1 mm) according to the specifications. You would find other mini-drill chucks with larger capacity.

I dislike the four-jaw configuration, but it's less expensive than a three-jaw chuck. It's easy to get a tiny drill bit misaligned when securing it in a four-jaw design.

My initial search appeared at first to return only four-jaw designs, but I was able to locate a three-jaw eventually:



It has a price similar to the Home Depot version, but appears to be an online resource known as iOffer.