Serving a S3 bucket and an API Gateway through CloudFront

I want to serve the front end of a web app from a S3 bucket at this url: www.acme.com.

At the same time, I wan't the back end be serve from an API Gateway stage (lambda functions are running behind) at this url: www.acme/api (I don't a api.acme.com url to avoid having to deal with a broken Same Origin policy).

At this time, I could set up Cloudfront to serve the S3 bucket or the API Gateway stage but not together.

I tried to declare both as ressources (first the bucket and then the API Gateway) and two paths in behavior (the default path to the bucket and a second /api to the Gateway). The bucket is served well but a GET to http://ift.tt/1QBKd2U leads to:

{"message":"Missing Authentication Token"}

Thanks for all clue about how I could resolve this issue.