The above answer is correct but I wanted to add some extra data as I too had the same requirement and I suspect most people using EB and RDS should have the same requirement too even if they don't know it yet.
First, why would you want the RDS instance to exist outside the EB environment? So that the lifetime of the RDS instance is not tied to the lifetime of the EB environment. Everybody should be doing this, and it was recommended during the recent AWS re:invent event.
But as implied but not explicitly stated above, when you do it this way you do not get the RDS_* variables automatically populated and need to retrieve their values and populate them yourselves via web console or .ebextensions (the latter not recommended for security reasons).
But then, the next problem is if you want to programmatically create environments (such as for blue-green zero downtime deployments) then you need a solution for how to populate the sensitive RDS values, such as password. Unfortunately, this requires you to drop further down the AWS stack and use a CloudFormation template.
The ideal solution is an enhancement to EB so that the "use an existing database" link mentioned in the question actually lets you manually associate an existing RDS database and then have the RDS_* environment variables automatically populated again, rather than redirecting you to unhelpful documentation. AWS Support said this has been raised as a feature request but of course no timeframe given.