Is there a way to setup a user to run SPs, but not access the underlying data tables? We have two database instances on two separate server boxes that are linked. The website user needs to be able to run stored procedures on DatabaseA which query DatabaseB tables, however it should not be able to read directly from tables on DatabaseB.

Is such a thing possible in Sql Server 2005?

Simply creating a Stored Procedure and giving the web user permissions to execute it returns an error of Access to the remote server is denied because no login-mapping exists.

I really don't want to give the website user account free access to DatabaseB tables.

Update

Aaron's suggestion of using EXECUTE AS OWNER does not work because Owner doesn't have access to DatabaseB. 

Using EXECUTE AS 'UserWithPermissions' returns an error of Access to the remote server is denied because the current security context is not trusted. Looking this up online leads me to believe this is because the database is not marked as TRUSTWORTHY, so the remote database rejects the connection. I suspect it would work fine if the two database instances were on the same server.

I did not want to set the web database as TRUSTWORTHY since some of the security on it is more relaxed, so I went with Aaron's other option: Create a User on DatabaseB that only has access to a few specific views containing the data the website should access, then go into the Security Properties on the Linked Server object and setup the website user to login as the new limited user when accessing the linked server.

This allowed my website user to have limited access to specific data in the private database, without giving it access to any of the underlying tables.
Couldn't you have procedure A run with execute as owner or execute as 'db user tied to security account associated with linked server'? That way the rights of the caller don't have to transfer...

Or, you can create a login on the remote server that only has the ability to execute the queries you want, and add an impersonation account to the existing linked server on Server A to have the local webserver login impersonate the new login on server B.