Definitely would be easier to answer with some sample data and expected outcome, but here's what I've come up with:

WITH newCustomers

AS (

SELECT

*,

COUNT(*) OVER (PARTITION BY CustomerID) AS orderCount

FROM

dbo.Orders AS O

WHERE

/* since PaymentDate is defined as DateTime using this is more consistent

method then between because with between you'd need to include the time

portion to be sure you get the full day for most recent date */

O.PaymentDate >= '2014-07-29' AND

O.PaymentDate < '2014-08-27'

)

SELECT

*

FROM

newCustomers AS NC

WHERE

/* the NOT EXISTS enforces that they have not had an order outside the

date range as well. I'm doing this because I interpreted this,

"but also taking into account if they have ever had a paid order outside of that date range",

to mean that you didn't want customers who had an order outside the date range. */

NOT EXISTS ( SELECT

1

FROM

Orders AS O

WHERE

NC.CustomerID = O.CustomerID AND

(

O.PaymentDate < '2014-07-29' OR

O.PaymentDate >= '2014-08-27'

) ) AND

NC.orderCount = 1;