Stored Procedure Security Question

Dear Group

I'm having two stored procedures, sp_a and sp_b

Content of stored procedure A:
CREATE PROCEDURE dbo.sp_a
SELECT * FROM a
GO

Content of stored procedure B:
CREATE PROCEDURE dbo.sp_b
SELECT * FROM b
GO

I have created a user that has execute permissions for both procedures.
When I run procedure A, all works fine but when running procedure B I'm
getting an error saying that the user must have SELECT permissions on table
B.

Both tables are owned by dbo, and the security role for the user doesn't has
any SELECT permission on table a and b.
I'd be grateful if anyone could point me in a direction why this error might
come up for procedure B but not for A,
with a possible solution without giving the user SELECT permissions.

Thanks very much for your help!

Martin