C# Get n distinct random numbers between two values whose sum is equal to a given number

I would like to find distinct random numbers within a range that sums up to given number.

Note: I found similar questions in stackoverflow, however they do not address exactly this problem (ie they do not consider a negative lowerLimit for the range).

If I wanted that the sum of my random number was equal to 1 I just generate the required random numbers, compute the sum and divided each of them by the sum; however here I need something a bit different; I will need my random numbers to add up to something different than 1 and still my random numbers must be within a given range.

Example: I need 30 distinct random numbers (non integers) between -50 and 50 where the sum of the 30 generated numbers must be equal to 300; I wrote the code below, however it will not work when n is much larger than the range (upperLimit - lowerLimit), the function could return numbers outside the range [lowerLimit - upperLimit]. Any help to improve the current solution?

static void Main(string[] args)
{
    var listWeights = GetRandomNumbersWithConstraints(30, 50, -50, 300);
}

private static List<double> GetRandomNumbersWithConstraints(int n, int upperLimit, int lowerLimit, int sum)
{
    if (upperLimit <= lowerLimit || n < 1)
        throw new ArgumentOutOfRangeException();

    Random rand = new Random(Guid.NewGuid().GetHashCode());
    List<double> weight = new List<double>();

    for (int k = 0; k < n; k++)
    {
        //multiply by rand.NextDouble() to avoid duplicates
        double temp = (double)rand.Next(lowerLimit, upperLimit) * rand.NextDouble();

        if (weight.Contains(temp))
            k--;
        else
            weight.Add(temp);
    }

    //divide each element by the sum
    weight = weight.ConvertAll<double>(x => x / weight.Sum());  //here the sum of my weight will be 1 

    return weight.ConvertAll<double>(x => x * sum);
}