You are asking for an answer that does not really exist, at least not with the degree of accuracy you seem to be looking for. Unless you know how the cup of flour in the recipe was measured out volumetrically, which is almost never stated, there is no way to know exactly what that cup of flour should weigh. There are just so many different ways of measuring out a cup of flour volumetrically and each method is likely to produce a different weight. Even if you know the brand of flour, there is no way to get an accurate conversion. If the recipe is a General Mills recipe, GM uses the Textbook method of measuring out a cup of flour volumetrically. King Arthur recommends the same method. That method is shown in the video at http://how2heroes.com/videos/dessert-and-baked-goods/bakers-tip-measuring-flour.
BTW, GM went to the Textbook method of flour measurement (they call it the "spoon & level" method) in October, 1989 (http://www.gmflour.com/gmflour/ourheritage.aspx
). Prior to that time, they recommended the "dip & level" method. If I had to choose between the two methods, I would say that most people still use the dip & level method, or some variation of it. Another point to keep in mind is that the Nutrition Facts for flours aren't exact, usually because of rounding factors.
The above said, one of our members, November, devised a tool that may be of use to you in respect of mass/volume conversions. It is called a Mass-Volume Conversion Calculator and it can be accessed at http://foodsim.unclesalmon.com/.
That tool works with only the ingredients, including flour, that are in the pull-down menu. In your case, you select the brand of flour from the pull-down menu and then select the flour Measurement Method that best or most closely describes the way you measure out--or wish to measure out--the flour volumetrically. Then enter the volume of flour that you want to convert to weight. Even with the Mass-Volume Conversion Calculator there is a human component involved in measuring out the flour and it will vary from one person to another no matter how careful that person is in measuring out the flour. So, you will never have a method that is "foolproof". If you play around with the tool with different flours and flour Measurement Methods, you will see how widely the answers can vary.
In case the particular flour that you are using is not in the pull-down menu, then I would look for the closest flour in the pull-down menu and use that as a proxy. The tool was devised using hundreds of actual weighings using different size measuring cups. To add a new flour would require that one go through the same drill. If you would like to see how I used November's tool recently to convert a volume recipe to weights, and particularly the flour, see the thread at http://www.pizzamaking.com/forum/index.php/topic,14442.msg144109.html#msg144109.
You will see that even when you can do a nominal conversion, you are not necessarily off the hook. You may still have to play around with weights to get the desired results.
EDIT (4/3/14): For the Wayback Machine version of the above GM link, see http://web.archive.org/web/20100105084108/http://www.gmflour.com/gmflour/ourheritage.aspx