I'm not an expert, but this is what little I know:

Are they blooming yet? Ants only like the flowers. If you support it so the flowers are off the ground it will help keep the ants from finding it too. I am trying out a tomato trellis (it's about 3 ft high) I found at kmart for $2 and it works well (you can see it on the "unknown peony pic" thread on the red peony.)
If it's done blooming then you don't have to worry about ants until next year, by then you'll have moved it anyway.

It's usually best to move/plant peonies in the fall, when they are dormant. They might have less flowers the first spring after you move them. Be sure to plant them shallow, buds facing up.
If you have a fence it's great to plant them against it, so that they will be supported. The blooms will always grow facing the sun, so plant them so that the fence will be between them and the sun, so they will lean on the fence (I hope that's clear). They will completely disappear during winter, so mark where you've put them. :)

Fall is also a great time to plant a lot of stuff, so it will be a good time to fill in the spot the peony is in now.
What zone are you in, btw?

First of all, though peony buds are attractive to ants, the ants are only interested in peony buds. They are not interested in invading you house. I have several peonies near all doors of my house and have never had ants inside. If you are worried just cut off the buds as they develop.

As ajpa said, move peonies in the fall when the foliage has turned brown.

Adding even 1/2" of soil on an established peony will result it in never blooming again. This may be the answer to your ant concerns.

-=beeky