Ben, I made the same mistake many years back, when I was first starting to use braid, and found the answer here on UB in discussions about reels. You can also use some mono type line as a backing direct to the reel spool and then attach the braid. Only needs to be several yards, just enough to get a good grip on the spool, but some folks will back the braid with more, since that requires less braid to fill the spool and mono lines are less costly.
Braid doesn't have any stretch so it won't tighten down on the spool like other lines do that have stretch. I know of a fellow who did the same thing with a big saltwater fly reel, and he was using a braided line as backing for his fly line since it's thinner than dacron backing and he would gain some extra backing capacity. He couldn't figure out why fish were still taking line. Most folks use an arbor knot to attach the line to the spool and with no stretch, the braid spins on the spool instead of the spool working against the drag. So a reel with a good drag, just stops the spool from spinning, but not the braid. I'll bet a lot of folks have had to figure that out eventually.