How do I avoid a useless return in a Java method?

As @BoristheSpider pointed out you can make sure the second return statement is semantically unreachable:

private static int oneRun(int range) {
    int[] rInt = new int[range+1]; // Stores the past sequence of ints.
    int count = 0;

    while (true) {
        rInt[count] = generator.nextInt(range); // Add randint to current iteration.
        for (int i = 0; i < count; i++) { // Check for past occurence and return if found.
            if (rInt[i] == rInt[count]) {
                return count;
            }
        }
        count++;
    }
}

Compiles & runs fine. And if you ever get an ArrayIndexOutOfBoundsException you'll know the implementation was semantically wrong, without having to explicitly throw anything.

The compiler's heuristics will never let you omit the last return. If you're sure it'll never be reached, I'd replace it with a throw to make the situation clear.

private static int oneRun(int range) {
    int[] rInt = new int[range+1]; // Stores the past sequence of ints.
    rInt[0] = generator.nextInt(range); // Inital random number.

    for (int count = 1; count <= range; count++) {
        ...
    }

    throw new AssertionError("unreachable code reached");
}