Generate uniform pseudo-random numbers in a closed interval

What's the best way to generate pseudo-random numbers in the closed interval [0,1] instead of the usual [0,1)? One idea I've came up with is to reject values in (1/2,1), then double the number. I wonder if there is a better method.

real x
do
   call random_number(x)
   if (x <= 0.5) exit
end do
x = 2*x
print *, x
end

The most important requirement is that the algorithm should not make a worse distribution (in terms of uniformity and correlation) than that generated by random_number(). Also I'd favour simplicity. A wrapper around random_number() would be perfectly good, I'm not looking to implement a whole new generator.

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As @francescalus points out in the comments, with the algorithm above lots of numbers in [0,1] will have zero probability of appearing. The following code implements a slightly different approach: the interval is enlarged a bit, then values in excess of 1 are cut out. It should behave better in that aspect.

real x
do
   call random_number(x)
   x = x*(1 + 1e-6)
   if (x <= 1.) exit
end do
print *, x
end