Hi dinalou ,
As the log says, you should check your error log and see the entries added there while synchronizing. If you can identify there the problem, please, tell us.
I see that you said your store is running out of memory, maybe this is the problem and you'll need to increase the max memory available. Can you tell us how much memory your server has available?
If you don't feel comfortable sharing this info here, you can open a private ticket.