Below is one of the suggested way in which i recently migrated my existing spring MVC web-application in Single page application using AngularJs as client side java-script framework:
1).First you needs to bisect your core business logic which must needs to handle in DB layer,if not done already. i.e. SP's,trigger's,etc...
2).Then for the thing mentioned as hardest by Toilal ("to migrate the existing server-side session management to the browser, because a REST API is stateless"), you can consider Spring Boot as one of the options to build things quickly instead of Spring MVC with fusion of java8 features to yield the JSON response via API's.
3).And than last but not the least, build UI layer with AngularJS in front End encapsulating the data returned by the spring boot API's. Please note that as angular is super power full, you can do most of your business logic at client side,your DB layer just needs to give you the whole bunch of business data to render the screen,other things you can easily handle in angular.
Thus, this way by making the DB layer thin and imbibing angular at front end, you can improve performance of your existing application drastically.
You can also check this AngularJS with Spring-mvc useful thread in same context.