Which AWS service will meet this requirement?

Explanations:

AWS App Mesh is primarily a service mesh that provides application-level networking to make it easy for services to communicate with each other. It is not specifically designed for publishing and managing REST APIs.

Amazon API Gateway is specifically designed to create, publish, maintain, monitor, and secure REST APIs at any scale. It allows developers to easily build and manage APIs for backend services.

Amazon CloudFront is a content delivery network (CDN) service that accelerates the delivery of static and dynamic web content but does not provide functionality for managing REST APIs directly.

AWS Cloud Map is a cloud resource discovery service that helps map resources in your applications but does not focus on publishing or managing REST APIs.