how to generate random numbers (probabilities) from exponential distribution that sum up to 1

Consider I want x random numbers that sum up to one and that distribution is exponential. When I use

x<-c(10,100,1000)

a<-rexp(x,rate=1)

a<-a/sum(a)

This will change the distribution, right?

So does anybody know a way so that the probabilities are still exponential distributed? I know that they will then not be completely independent anymore.

Thanks a lot!