To find out if it is a hardware or a software problem you can :
Boot from an external USB disk or Superdrive, if the internal speakers work again than there is no hardware problem on your MBAir.
If you do not have a bootable ext. Disk at hand : remove all cables/devices except the power cord.
Restart your MacBook Air while holding down the "D" key.
In Apple Hardware Test menu, follow the instructions.
Info about the hardware test can be found here : http://support.apple.com/kb/HT2644
When the Apple Hardware Test finds no problems, something went wrong with the software.
When both test say it's hardware... Hope your warrenty is still valid.
If it's software, let us know.
Note : this tip assumes audio is working when booted from an ext. HD or other bootable partition.
When hardware test is OK and you have the latest Software Update but still do not see an device (input or output) in the utility Audio MIDI Setup.app, try :
delete the 2 audio preference files in HD / Library / Preferences / Audio /
restart the computer
Still not working ?
Re-install OS X from the installation disk and run Software Update....
Still not working ?
I'm out of options...ask your local dealer or try Apple Support Express Lane http://www.apple.com/support/
Though there could be a variety of causes, there may be a hardware issue.
There is a long discussion of this issue here (with several suggestions): "MacBook Air no audio output device found ???????? *please help* " (https://discussions.apple.com/thread/1736638?start=285&tstart=0)
My original Macbook Air also has this issue and I believe that my internal audio cable has failed. This failure has been experienced by a number of people with the original/early Macbook Air.
Some people have had success (even out of warranty) in having it fixed by Apple. (No luck for me)
There is a guide to replacing the cable yourself here:
I had the same problem on my MacBook. The sound just stoped working. I brought it to the apple store and they did a fresh install. After the sound worked with the headphones but not the internal speakers. so they replaced the logic board ($400). It all worked good until I restored from my time machine back up. My backup was from after the sound stoped working, so I tried one from before the sound issue. Magically it worked again. I then just restored my recent files individually (not OSX again)
About a week ago it stoped again. I restored from a backup and it's worked fine since.
So before you go to the apple store to get it fixed, if you have a backup try restoring to it. Then just copying your files back. If you don't have a backup you could try doing a clean install of the latest version of OSX, then putting your flies back on. The only thing with doing a clean install is you would lose your preferences (you wouldnt if you restored from a full time machine backup)