So, if I understand you correctly, you are trying to say:
- You have a MultiBit wallet with the address 1DPYi3JbPwwE6inR3kBBQMx2u4k5kHxSwv
- You also have a blockchain.info wallet with the address 1DPYi3JbPwwE6inR3kBBQMx2u4k5kHxSwv
- On 2014-05-06 at 23:08:06 UTC you received 0.0348 BTC at 1DPYi3JbPwwE6inR3kBBQMx2u4k5kHxSwv
- You confirmed that the bitcoins were received in the MultiBit wallet
- You then shut down the MultiBit wallet
- On 2014-05-07 at 19:30:49 UTC while your MultiBit wallet was shut down, the 0.0348 BTC were sent to 17hghQzNMXkCZw8Dj6HPEYAXnYZgAP99RL
- You did not create the transaction sending the bitcoins to 17hghQzNMXkCZw8Dj6HPEYAXnYZgAP99RL and you are not the owner of the address
It sounds to me like someone else has your private key from the 1DPYi3JbPwwE6inR3kBBQMx2u4k5kHxSwv address.
Since you have the same address in at least two different wallets, I guess the first question would be:
Where did the 1DPYi3JbPwwE6inR3kBBQMx2u4k5kHxSwv address come from? Was it created in MultiBit and imported into blockchain.info? Was it created in blockchain.info and imported into MultiBit? Was it created somewhere else entirely, and then imported into both MultiBit and blockchain.info?
If you obtained the private key from an insecure source, then that was your most likely leak.
If you exported the private key and then imported it into another wallet, then there is a good chance that you weren't careful with the private key after exporting it.
If you imported the private key into any other "Wallets" it is also possible that one of them were compromised.