Partially-Unique Check Constraints
You can use a filtered unique index for this:

CREATE UNIQUE INDEX LinkedID_Unique_except_Zeroes_or_Null 
    ON dbo.oldAccessTable (LinkedID)  
    WHERE LinkedID &lt;&gt; '00000000' ;  


The condition (LinkedID &lt;&gt; '00000000') takes care of both requirements, i.e. rows with '00000000' or NULL are ignored and not stored in the index, so they are not checked for uniqueness.