Easiest solution. Tell your client to change his password and give you the new password. Publish the app on iTunes, tell your client to change his password back. If he doesn't trust you enough to give you a half-hour access to his account, why on earth does he trust you to write the app in the first place.
Next easiest solution (assumes you have a paid up account.) You publish the app on your account and mark it so it doesn't automatically go in the store after approval. Once the app is approved, transfer ownership to him. There are a whole bunch of restrictions on this process, including No iCloud, Passbook or in-app purchase subscriptions. For more details, Google "Transferring and Deleting iOS Apps" or see [https://developer.apple.com/library/ios/documentation/LanguagesUtilities/Conceptual/iTunesConnect_Guide/Chapters/TransferringAndDeletingApps.html]
Next easiest solution would be to give the customer access to the source and explain to him how to build the app and put it in the store.
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A new solution is available now. The client could give you admin privileges to his appstore (https://appstoreconnect.apple.com) account.