  Background: The rate \lambda_pp\mu\ characterizes the formation of pp\mu\
molecules in collisions of muonic p\mu\ atoms with hydrogen. In measurements of
the basic weak muon capture reaction on the proton to determine the
pseudoscalar coupling g_P, capture occurs from both atomic and molecular
states. Thus knowledge of \lambda_pp\mu\ is required for a correct
interpretation of these experiments.
  Purpose: Recently the MuCap experiment has measured the capture rate
\Lambda_S from the singlet p\mu\ atom, employing a low density active target to
suppress pp\mu\ formation (PRL 110, 12504 (2013)). Nevertheless, given the
unprecedented precision of this experiment, the existing experimental knowledge
in \lambda_pp\mu\ had to be improved.
  Method: The MuCap experiment derived the weak capture rate from the muon
disappearance rate in ultra-pure hydrogen. By doping the hydrogen with 20 ppm
of argon, a competing process to pp\mu\ formation was introduced, which allowed
the extraction of \lambda_pp\mu\ from the observed time distribution of decay
electrons.
  Results: The pp\mu\ formation rate was measured as \lambda_pp\mu = (2.01 +-
0.06(stat) +- 0.03(sys)) 10^6 s^-1. This result updates the \lambda_pp\mu\
value used in the above mentioned MuCap publication.
  Conclusions: The 2.5x higher precision compared to earlier experiments and
the fact that the measurement was performed at nearly identical conditions to
the main data taking, reduces the uncertainty induced by \lambda_pp\mu\ to a
minor contribution to the overall uncertainty of \Lambda_S and g_P, as
determined in MuCap. Our final value for \lambda_pp\mu\ shifts \Lambda_S and
g_P by less than one tenth of their respective uncertainties compared to our
results published earlier.
