Let $X$ ~ Uniform(0,1) with pdf $f(x)$:

<img src="http://www.tri.org.au/se/Uniformpdf01.png">

The joint pdf of the 1st and $n$th order statistics is, say, $g(x_\left(1\right), x_\left(n\right))$:

<img src="http://www.tri.org.au/se/OrderstatUniformpdf01.png">

where I am using the `OrderStat` function from the `mathStatica` package for _Mathematica_ to automate the nitty gritties for me (I am one of the authors of the former).

... and with domain of support:

<img src="http://www.tri.org.au/se/domainOrderstatUniformpdf01.png">

We seek $P(X_\left(n\right)- X_\left(1\right) > d)$:

<img src="http://www.tri.org.au/se/proborderstatsol.png">

All done.

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Just as a side note, the solution obtained here is different to that of Carlo's posting:

<img src="http://www.tri.org.au/se/carloresult.png">

whereas I obtain, for the same parameter values $\frac{8}{9}$ [Checked via Monte Carlo ]