There is no built-in way to terminate services based on budgets or billing alarms.
You can get notified automatically, but it is then up to you to determine how to handle it.
Would you really want AWS automatically terminating your production infrastructure because you went $1 over your estimated monthly spending?
Edit: There is now a way to monitor and alert on free tier usage, and when your predicted usage will exceed the free tier. See here for details. You could probably come up with a way to terminate infrastructure based on an alert using SNS & lambda.
Edit 2: In Oct. 2020, AWS released Budget Actions – the ability to trigger an action when a budget thresholds are reached. This should give you the ability to automate a response – you can shut down servers, change IAM permissions to prevent additional infrastructure from being created, etc.