Briefly: 5 year old MBA 1,1; 'genius' at the apple store told me that it would be $300 for a new logic board; I consulted the forums instead and ordered an audio cable through ifixit.com; thus, I fixed it myself and I heard the chimes singing on boot-up.
I can see responses have varied for this thread, and people have probably experienced a number of problems that invariably manifested as "no audio output device." So i can only tell you my story and what worked.
I have a Macbook Air 1,1 (first generation, first model) that did not have too many problems while under warranty. Then just before hitting our 5 year anniversary my MBAs audio output became unreliable and then entirely non-functioning.
I tried different round-about fixes listed in the forums to see if it was a software problem, and even took it into to see an apple genius. The genius hooked his machine up to mine and confirmed that it was a hardware problem, and then recommended that it would need a new logic board. With a smug grin he said he could replace it for $300. I recalled reading about a replacement audio cable as mentioned in the forums, and he said he could replace that for $50 even though I could see that it was listed for $10 in his inventory catalogue.
Because my machine was outside or warranty, I ordered the cable myself for the cheapest, easiest fix. I opened up the under-belly of my MBA as an amateur tinkerer, and made the fix in an easy 30 minutes, I also installed a solid state drive. I turned on my machine to hear again the glorious chimes of the boot-up.
