Update CHANGELOG.md

- [ ] Ensure the CHANGELOG is up to date with all relevant commits since the last release
- [ ] Add the version number for this release & the date to the CHANGELOG, underneath "## Unreleased" 
  - e.g. "## 1.2.3 - 2022-02-14"
- [ ] Update the README if necessary (this is only required when there are breaking changes in the release, such as dropping support for an iOS || Android version)

CHANGELOG.md

@@ -2,6 +2,8 @@
 
 ## Unreleased
 
+## 0.39.0 - 2024-10-15
+
 **Features**
 
 - Adds support for CustomerSession in private beta [1744](https://github.com/stripe/stripe-react-native/pull/1744)