This question will come as a contradiction first..."exchange" and "safest" don't go to together.

That said, the safety that comes with a Bitcoin (or other alt crypto-currency) wallate comes with how easily anyone else can open it. That ease comes from a very few specific criteria; 1. Does someone other than you have access to your wallet's private keys? (every online wallet does) 2. Is your wallet connected to the internet? (if so it is vulnerable to attack and hack).

The only way to truly get around these risks is to have a wallet that lives off line and gives you complete and sole control to the private keys.

Trezor comes to mind first, but the aren't the only good option. Read about.