For this weave, I used 5mm and 7mm rings. They seemed to work perfectly. I think you could also use the 6mm and 8mm rings, but I haven't tried those yet. This is also a very dense weave and it does take quite a few rings. I use the pre-closed small ring method. It goes faster if you close a bunch of small rings and open a bunch of large rings, so they are all ready to go.

Step 1. Make a chain like the one shown above in the desired width.

Step 2. Now make another chain, starting with a small ring that is one large ring shorter than your first chain.  End with a small ring. The bottom chain is the first chain and the top chain is this step.

Step 3. Lay the second chain on top of the first, so that the large rings lay over the small rings. 

Step 4. To add the next row. Pass a large ring through the 2nd small ring and around the first small ring, add 1 small ring.  

Step 5. Pass a large ring through the fourth small ring and back through the second small ring and the small ring you just added. Add a small ring.   

To finish the row, add rings like you did in step 5. When you add your last large ring, you will only pass through the 2 small rings to the left of it, and don't add any small rings. 

Step 6. For the next row, pass a large ring through the first and third small ring, add 2 small rings. 

To finish the row, pass a large ring through the third small ring and the small ring you just added. Add a small ring and pass through the fifth small ring. Continue to the end of the row.

Just keep repeating steps 5 and 6 until you have the desired length. If you get confused, check out www.mailleartisans.org.  
There are a couple of tutorials there that might help you.