Users regularly enter sensitive data, such as passwords, credit card numbers, or tax information, into the browser window. While modern browsers provide powerful client-side privacy measures to protect this data, none of these defenses prevent a browser compromised by malware from stealing it. In this work, we present Fidelius, a new architecture that uses trusted hardware enclaves integrated into the browser to enable protection of user secrets during web browsing sessions, even if the entire underlying browser and OS are fully controlled by a malicious attacker. Fidelius solves many challenges involved in providing protection for browsers in a fully malicious environment, offering support for integrity and privacy for form data, JavaScript execution, XMLHttpRequests, and protected web storage, while minimizing the TCB. Moreover, interactions between the enclave and the browser, the keyboard, and the display all require new protocols, each with their own security considerations. Finally, Fidelius takes into account UI considerations to ensure a consistent and simple interface for both developers and users. As part of this project, we develop the first open source system that provides a trusted path from input and output peripherals to a hardware enclave with no reliance on additional hypervisor security assumptions. These components may be of independent interest and useful to future projects. We implement and evaluate Fidelius to measure its performance overhead, finding that Fidelius imposes acceptable overhead on page load and user interaction for secured pages and has no impact on pages and page components that do not use its enhanced security features.