Try using a waste block about 2/3rds the diameter of the plate or so and use double sided tape.  Keeping the tailstock up for support.  just makes sure you mark a good accurate center point on the back side before you mount it on your intial waste block.  
    Actually, now that I think about it, Here is what I might do.  If the piece is square to begin with, I would mark centers on both sides and rough cut it round on my bandsaw.  I say rough cut but you will want to get it as close to the size you want and still be able to spin it on the lathe.  Next I would glue on a waste block but use the lath as my clamp between centers (there is a reason to use the drive and live center) and let it dry the way.
    After the glue has dried a bit go ahead and finish rounding the piece.  Round the waste block as well to a size that will fit in whatever chuck you want to use.  Let the glue dry completely and the mount the piece in your chuck for finishing the face/front/top of the plate.  
    When done with that, remove the plate from the chuck.  This next step can be done however you want.  I would use a faceplate and make me another wasteblock out of a scrap piece of 3/4" plywood and make sure my plate top/front/face will fit on it completely and attach it to my faceplate, make sure it is smooth flat and round and cover that with double sided tape.  
     Remember when I said that there was a reason for using the drive center and live center earlier, well here is why.  To make the plate center much easier for turning the backside, move the tail stock up fairly close to the and place the back side of the plate (with the wasteblock still attached) onto the tailstock live center point.
    The dimple made by the drive center will allow you to more accurately center the plate.  Then using the handwheel on the tail stock gently move the plate towards the headstock and press it against the waste block with all the tape on it.  Keep the tail stock in place as long as possible.  When you eventually have to move it, the tape should be able to hold long enough to finish off the bottom nicely.
    You may have to do a bit of touch up sanding on the face of the plate but I don'tthink it will be too bad.
  If you decide to try it let me know how it works.

Curt, you said that you want the bottom to have a recess.  Glue your waste block to what will be the top, turn the bottom including the recess that just happens to be of such a size that you can mount it on your chuck in expansion mode.  Problem solved.  Do your sanding, flip it over, turn the top, sand it.  Lay the finish down and your done. 

That's one way.   8)

This is my method which doesn't use a waste block. At least its the method I was taught.
Mount your stock between centers. Turn a tendon on the tail stock side. This will be the top of the plate. You don't need to shape it just yet all you want is a tendon for your chuck. Once the tendon is cut, attach your chuck and reverse the plate griping the tendon you just cut. Your now going to shape the bottom of your plate. Make the shape you want making the bottom flat. Here is where you will want to cut a recess tendon ( sorry I forgot the correct term, it's a recess that you cut so you can use your chuck and expand the jaws so they are press outwards against the recess you just cut). I have heard a recess of only a 32nd but I have cut my a little deeper. You'll be surprised how well it will hold.  Once your outside or bottom of the plate is shaped, reverse the plate again, this time you will finish the top of the plate. Remember to take light cuts since you will only be holding the plate using the jaws in a expanding mode.  Continue with your cuts until you have the desired shape. You can use cole jaws to clean up the bottom or just sand it by hand. Apply your finish and your done.  I hope that is clear if not I can take some photos and post them if you want.

I see Rev was posting while I was typing this...

That's what keep it fun...  and Yes I have gone through the bottom of several

That is similiar to how I have done small trivits and coasters. I actually used a screw center, but only about a 1/4 inch deep, then turned the bottom recess.