you should be able to use your old iPhone without SIM card, as if it were like an iPod touch
I did this with my 3G when I first got my 3GS - for a while my kids were using the old 3G with all my same apps only just using WiFi, while I used the original SIM card in my new 3GS (I've since sold the old phone).
the key is that the old phone must be left activated. In other words, if you do a complete restore of the old phone to wipe it clean, it will not even operate as an iPod until you activate it with your SIM card (like you did when you first got it). but after that it's ok to use without SIM card.
*EDIT* - I just found an Apple support document that shows a difference between original iPhone and later models regarding this issue.
according to the doc, the 3G and 3GS will operate without a SIM as I have described above, i.e. once activated you can remove the SIM card and it will work as if it were an iPod touch
However the original iPhone (which is what the OP has) behaves differently.
it says to use the iPhone without active cellular you need to have the original SIM card it was activated with installed *even if the SIM card is now inactive*
so I think a solution (without jailbreaking) for your original iPhone would be to somehow get AT&T to switch your cellular account to a new SIM card (your dog ate your old SIM card, perhaps?), then leave the old now-inactive SIM card in the original iPhone? unfortunately this would not be a zero-cost method...