Find the missing and duplicate elements in an array in linear time and constant space

Question

You’re given an array of N 64 bit integers. N may be very large. You know that every integer 1..N appears once in the array, except there is one integer missing and one integer duplicated.

Write a linear time algorithm to find the missing and duplicated numbers. Further, your algorithm should run in small constant space and leave the array untouched.

Source: http://maxschireson.com/2011/04/23/want-a-job-working-on-mongodb-your-first-online-interview-is-in-this-post/