There is growing interest in providing programmatic access to the value locked in Bitcoin, which famously offers limited programmability itself. Various approaches have been put forth in recent years, with the vast majority of proposed mechanisms either building new functionality on top of Bitcoin or leveraging a bridging mechanism to enable smart contracts that make use of "wrapped" bitcoins on entirely different platforms.In this work, an architecture is presented that follows a different approach. The architecture enables the execution of Turing-complete Bitcoin smart contracts on the Internet Computer (IC), a blockchain platform for hosting and executing decentralized applications. Instead of using a bridge, IC and Bitcoin nodes interact directly, eliminating potential security risks that the use of a bridge entails. This integration requires novel concepts, in particular to reconcile the probabilistic nature of Bitcoin with the irreversibility of finalized state changes on the IC, which may be of independent interest.In addition to the presentation of the architecture, we provide evaluation results based on measurements of the Bitcoin integration running on mainnet. The evaluation results demonstrate that, with finalization in a few seconds and low execution costs, this integration enables complex Bitcoin-based decentralized applications that were not practically feasible or economically viable before.