Is there a solid "Bitcoin Bank" yet? Once MyBitcoin was thought to be a "solid bitcoin bank". Then it was hacked and a lot of people lost a lot of money there.

Is there another, secure alternative for storing Bitcoins except storing them on your own computer?

(Obviously correctly storing Bitcoin on your computer, with proper encryption and backup, will always be more secure because you don't have to trust anyone. Still, for a lot of people, an online Bitcoin bank is a must.)
Currently, the most secure place to store bitcoins online appears to be the exchanges. There is some evidence that Mt. Gox is also hoping to become the default bitcoin bank as they are developing mobile payment applications for instant payment.

http://mtgoxlive.com/mobile/

They have also reimbursed the losses of another bitcoin exchange.

https://www.mtgox.com/press_release_20110811.html

It is worth pointing out that while Mt Gox was hacked, no one lost any bitcoins that were not reimbursed.