Updated answer for Dec 17 - since lots of answers on this thread recommended services that have since seized to operate.
The most secure way to mix Bitcoin is - from experience - to:
1. Exchange the BC to XMR Monero first, by whatever exchange you like - I recommend
https://shapeshift.io(Monero uses different, more private technology and most XMR wallets have a mixing parameter by default when sending!)
2. then back to bitcoin via
https://xmr.to - which also inherently mixes the XMR with other XMR then sends you fresh Bitcoin instantly.
Xmr.to is widely used and entirely reliable. Very straightfoward too. Unlike big exchanges, you don't need to register or even use your email. Just amount of bitcoin you want + the address you want it sent to. Then you get instructions and options of payment by monero. They website sends you the bitcoin so quickly it's very refreshing to see after working with major exchanges.
If they have any issue with their server - the site won't let you proceed with the order and let you know it's offline. When it is offline, it's back up in seconds once you refresh.
All in all, I would say an exemplary service.