The Rust type system guarantees memory safety and data-race freedom. However, to satisfy Rust's type rules, many familiar implementation patterns must be adapted substantially. These necessary adaptations complicate programming and might hinder language adoption. In this paper, we demonstrate that, in contrast to manual programming, automatic synthesis is not complicated by Rust's type system, but rather benefits in two major ways. First, a Rust synthesizer can get away with significantly simpler specifications. While in more traditional imperative languages, synthesizers often require lengthy annotations in a complex logic to describe the shape of data structures, aliasing, and potential side effects, in Rust, all this information can be inferred from the types, letting the user focus on specifying functional properties using a slight extension of Rust expressions. Second, the Rust type system reduces the search space for synthesis, which improves performance. In this work, we present the first approach to automatically synthesizing correct-by-construction programs in safe Rust. The key ingredient of our synthesis procedure is Synthetic Ownership Logic, a new program logic for deriving programs that are guaranteed to satisfy both a user-provided functional specification and, importantly, Rust's intricate type system. We implement this logic in a new tool called RusSOL. Our evaluation shows the effectiveness of RusSOL, both in terms of annotation burden and performance, in synthesizing provably correct solutions to common problems faced by new Rust developers.