There are no Database constructors that take an actual connection. The design philosophy of the block is to manage the connections for you.
You can get a connection using the CreateConnection() method or retrieve a connection string, modify it and create a new Database object. However, for most of the methods the block will manage the connections internally anyway so I'm not sure that
will buy anything.
Perhaps the best approach would be to retrieve the connection string from the existing Database object, modify it (you can use one of the
DbConnectionStringBuilder classes to do this quite easily) and from there use a straight ADO.NET approach which would allow you to manage all aspects of connection lifetime.
Enterprise Library support engineer