I would like to say something strikingly different from those ultra-optimistic opinions listed above (mostly by US employees, who I feel are to biased to give you good advice!)
I think you do have reason to be afraid. "Obvisouly you can tell there is no reason to fear flying U." I disagree- Obviously, there is MORE reason to fear flying U than there is British or Air France or some airline like that, which is in no danger of not existing anytime soon.
Having said that, it looks like they are doing a little better than anyone expected. And, the likelihood of them not beign around in July is low- I think they will last longer than that even if they do eventually fold.
I flew to Rome on one of their A330s, and really did like it. FAR
better than my longhaul experience on American (US even had soap and toilet tissue is the lavatories... fancy that!)
So, my advice, in spite of realizing and agreeing with your concern, is to go ahead and fly US. Nice planes, service was fine for coach (check out the lumbar button on your seat!) and in this case, a great price.
But I would suggest, like Outboy, that you pay with a credit card. I booked a trip on Legend with my Amex, and when they went bust, Amex had no trouble with it and did not charge me anything. And, in the event of failure, US will probably book you on another airline. Legend was booking folks on Delta. i would expect US to book them on United or another.
So- go for it, but pay with a card!