LLM-integrated app systems extend the utility of Large Language Models (LLMs) with third-party apps that are invoked by a system LLM using interleaved planning and execution phases to answer user queries. These systems introduce new attack vectors where malicious apps can cause integrity violation of planning or execution, availability breakdown, or privacy compromise during execution. In this work, we identify new attacks impacting the integrity of planning, as well as the integrity and availability of execution in LLM-integrated apps, and demonstrate them against IsolateGPT, a recent solution designed to mitigate attacks from malicious apps. We propose Abstract-Concrete-Execute (ACE), a new secure architecture for LLM-integrated app systems that provides security guarantees for system planning and execution. Specifically, ACE decouples planning into two phases by first creating an abstract execution plan using only trusted information, and then mapping the abstract plan to a concrete plan using installed system apps. We verify that the plans generated by our system satisfy user-specified secure information flow constraints via static analysis on the structured plan output. During execution, ACE enforces data and capability barriers between apps, and ensures that the execution is conducted according to the trusted abstract plan. We show experimentally that ACE is secure against attacks from the InjecAgent and Agent Security Bench benchmarks for indirect prompt injection, and our newly introduced attacks. We also evaluate the utility of ACE in realistic environments, using the Tool Usage suite from the LangChain benchmark. Our architecture represents a significant advancement towards hardening LLM-based systems using system security principles.