Nevyn changed my create table statements to use your table names and ran your code:

SELECT *

FROM Table_A a

LEFT OUTER JOIN

(SELECT B_ID,A_ID,City,State,Zip

FROM Table_B b

INNER JOIN

(SELECT A_ID,bid=(min B_ID)

FROM Table_B

GROUP BY A_ID) as inner

ON inner.bid=b.B_ID) as outer

ON a.A_ID=outer.A_ID

Result: Msg 102, Level 15, State 1, Line 7

Incorrect syntax near 'B_ID'.

Wondering what I did incorrectly?