How to generate a random number that have two constraints (mean and range)?

I try to generate N numbers of random number from 0 to 1. This can be achieved via the rand function, but how to constraint the mean of all the N numbers is 1? The following code will generate a unit mean, but it has exceeded (<0 or >1) the range I wanted (0<pp<1).

N=100;
p=rand(N,1);
pp=p-(mean(p)+1)

Thanks in advance!