Select Fields Based On The Value In A Field

I have a table with two sets of fields, one for PRIMARY Mail Address and the other for ALTERNATE Mail Address. Both sets are five fields long.  I will call them PRIMARY & ALTERNATE for this discussion.  I want to select the PRIMARY fields if ALTERNATE is either null or zero, and the ALTERNATE fields if data is present.

I have used this syntax but it gives a table with both the PRIMARY & SECONDARY records.

SELECT    TXPRCL, TXALTR, TXANAM
FROM    dbo.PCWEBF21
WHERE    (TXALTR > 0)
UNION
SELECT    TXPRCL, TXTAXP, TXTNAM
FROM    dbo.PCWEBF21
WHERE    TXALTR = 0

Would a Select Case work? something like

Select PRIMARY when ALTERNATE = 0, and ALTERNATE when ALTERNATE > 0 or is not null.