Here is the RoutingRule I ended up with:
<RoutingRules>
<RoutingRule>
<Condition>
<KeyPrefixEquals>api/</KeyPrefixEquals>
</Condition>
<Redirect>
<HostName>api.example.com</HostName>
<ReplaceKeyPrefixWith/>
</Redirect>
</RoutingRule>
</RoutingRules>