Could miners steal money? I've been reading around: https://en.bitcoin.it/wiki/How_bitcoin_works

What prevents a miner's block from being accepted across the network if the hash checks out when miner forges transactions in the block?

For example, lets say I'm mining and I want to steal money from address A and put it into my address B. So I build a block with the A->B transaction along with some requested transactions. Now if I can compute a hash for the block before anyone else, and send off to peers in the network, how could they reject it? How would they know that's not a real transaction? If bitcoin clients record requests to verify, maybe this request was blocked by some NAT issue.

I can't think of a good way of fixing this and I didn't find anything about in documentation. Any insight would be appreciated. Thanks.
Transactions are signed with ECDSA. Incidentally, bitcoin addresses are hashes of the ECDSA public key that can spend them. You can't spend somebody else's coins unless you 1) do a bruteforce search to find their private key or 2) find a public key that hashes out to the same thing. Both are much, much harder than doing a double spend attack.