With the launch of the Chandra X-Ray Observatory and XMM-Newton , X-ray astronomy has entered an era of precision science .
The precision observational data taken by these satellites require precision basic physics data for their analysis .
The present authors have recently published a series of papers in which they have reported the results of the accurate calculations of the electron-ion thermal bremsstrahlung which are particularly suited for the analysis of the high-temperature galaxy clusters .
In this paper we will present accurate analytic fitting formulae for the nonrelativistic electron-electron thermal bremsstrahlung Gaunt factor which will be useful for the analysis of the X-ray observational data taken by the Chandra X-Ray Observatory and XMM-Newton .
With the publication of the present paper X-ray astronomers will be provided with an accurate thermal bremsstrahlung Gaunt factor which has an overall accuracy better than 1 % for the temperature range 6.0 \leq log T [ K ] \leq 8.5 .