This Letter describes new work on the determination of the Newtonian constant of gravitation, G, carried out at the BIPM since publication of the first results in 2001. The apparatus has been completely rebuilt and extensive tests carried out on the key parameters needed to produce a new value for G. The basic principles of the experiment remain the same, namely a torsion balance suspended from a wide, thin Cu-Be strip with two modes of operation, free deflection (Cavendish) and electrostatic servo control. The result from the new work is: G=6.67545(18)×10-11m3kg-1s-2 with a standard uncertainty of 27 ppm. This is 21 ppm below our 2001 result but 241 ppm above The CODATA 2010 value, which has an assigned uncertainty of 120 ppm. This confirms the discrepancy of our results with the CODATA value and highlights the wide divergence that now exists in recent values of G. The many changes made to the apparatus lead to the formal correlation between our two results being close to zero. Being statistically independent and statistically consistent, the two results taken together provide a unique contribution to determinations of G.From here citing "Improved Determination of G Using Two Methods", Phys. Rev. Lett. 111, 101102 (2013) link.aps.org/doi/10.1103/PhysRevLett.111.101102/
Simply put, the accuracy with which we have measured gravity isn't all that great. The CODATA result has four significant digits, and this latest measurement claims five. The cosmological constant, which is the only other constant of general relativity, is known to only a one significant digit accuracy.
We also haven't made much progress. The Cavendish experiment in 1873 measured G to three significant digits as 6.74*10-11, which we now know had an error of about 1%. Today, 140 years later, despite immense technological advances, the precision of our measurement of this constant has improved by a mere factor of 1,000 to 10,000, depending on who is giving the answer.
Of course, there is good reason for this as measuring the relatively weak gravitational force in a precisely controlled experimental setting is very hard.
We can measure gravitational effects such as those that determine the location of the Moon relative to the Earth with far greater precision (a few millimeters at that distance), but measuring the absolute value of the constant is much harder because it is hard to determine the exact mass of bodies like the Moon and the Earth. When we measure gravitational effects in something like the Earth-Moon system, the imprecision in the absolute mass of the Moon, the absolute mass of the Earth, and the absolute value of the gravitational constant can be made irrelevant by calibrating the experiment with a measurement that depends simultaneously upon all three of these quantities and can measure general relativity corrections that have a precision of 1013 or so.
We also have no accepted theoretical formula of other more easily measured constants that can be used to indirectly determine the value of G. In contrast, the speed of light, for example, could be determined using the measured value of two other more easily measured values in Maxwell's Equations that provided great precision relative to direct measurements. The current experimental measurement is also sufficiently imprecise that it can't discriminate between any number of theoretical proposed ways to derive G from other physical constants. Coming up with many different ways of generating a four digit number from known mathematical and physical constants is quite easy, so one needs more than a numerical match to recommend a proposed theory for deriving G from other more easily measured constants.
Still, how bad is the problem? Physicists were complaining in public about the lack of precision with which this constant was measured in 1898 and have really never stopped. As an agenda for a conference set for February 27, 2014 explains:
The Newtonian constant of gravitation, G, is the only fundamental constant of physics for which the uncertainty given in successive CODATA evaluations has increased. Measurements made since 2000, using a variety of methods, now show a spread of values more than ten times their estimated uncertainties. We shall explore possible reasons for this and hope to come to some proposals for new measurements that might resolve the present impasse.While there are some Standard Model constants that we know only to comparable levels of precision (e.g. the quark masses and some of the PMNS matrix angles), the Standard Model electroweak constants are mostly known to vastly greater precision and the Standard Model has been much more rigorously tested in a wide variety of circumstances than General Relativity (which is not to say that there is any evidence whatsoever that General Relativity is not empirically correct).