rjv140, yes adding a stock to that pistol would turn it into a short-barreled rifle. The process David described is what you would do if you want to register the SBR as an individual.
I don't know your exact situation, but it's important to bear in mind, however, that individual registration brings with it a few limitations that may or may not be of concern for you. If you register the SBR as an individual, ONLY YOU can legally have access to the rifle. What does that mean, exactly? Well, you can let other people handle or shoot the rifle as long as you are present, but no one else may have access to the gun in your absence. Say, for example, that you are married and have a gun safe. If you lock the SBR in the safe, your wife cannot legally know the combination to the safe because that would give her access to the gun when you are not present. Another drawback of the individual registration route is that you have to get a CLEO sign-off (which they are not required to do, and furthermore cannot be compelled to do). If your local CLEO will sign, consider yourself lucky. Many will not.
The way most people avoid the above problems is that they create a trust and transfer the NFA items to it as trust property. This is by far the best current method of obtaining NFA goodies, because it allows for anyone who is a trustee in the trust to have access to the NFA items. The trustees can freely transfer the items among themselves. If there is anyone that you would like to have access to the rifle while you are not present, the trust is the way to go. Also, if you go the trust route you are not required to get a CLEO signature, submit passport photos, or prints.
To answer your question about waiting until you have the gun in your possession before sending in the NFA paperwork, you need to have the pistol that you want to SBR in hand so that you can list its serial number, and other relevant info on the paperwork that you send in.
Also, the 6-8 month wait time should be going down to a more reasonable level very soon as ATF has brought on assistants to help with paperwork. Don't let the wait time scare you off! Owning NFA stuff is easier than you'd think it would be.
"Enjoy every sandwich." -- Warren Zevon