This should be enough:
double v = Math.Round(myRandom.NextDouble(), 10);
The difference between 0.123
and 0.1230000000
is a matter of formatting, see the answer from @SamIAm.
After the Edit:
double next = r.Next(1000000000, 9999999999);
return double.Parse(String.Format("0.{0}", next));
this is getting an integer between 1000000000 and 9999999999 and then uses the default culture to convert it to a double (in the 0 ... 1.0 range).
Since you seem to use the comma (,
) as a decimal separator, at least use
return double.Parse(String.Format("0.{0}", next), CultureInfo.Invariant);