GRB 210510***A***: Swift-BAT refined analysis S. D. Barthelmy (GSFC), J. R. Cummings (CPI), H. A. Krimm (NSF), S. Laha (GSFC/UMBC), A. Y. Lien (GSFC/UMBC), C. B. Markwardt (GSFC), D. M. Palmer (LANL), T. Sakamoto (AGU), M. Stamatikos (OSU), T. N. Ukwatta (LANL) (i.e. the Swift-BAT team): Using the data set from T-239 to T+963 sec from the recent telemetry downlink, we report further analysis of BAT GRB 210510***A*** (trigger #1047857) (NAME, et al., GCN Circ. xxxx). The BAT ground-calculated position is RA, Dec = 255.983, -37.861 deg which is RA(J2000) = 17h 03m 55.8s Dec(J2000) = -37d 51' 41.2" with an uncertainty of 1.8 arcmin, (radius, sys+stat, 90% containment). The partial coding was 100%. [Description about the mask-weighted light curve] T90 (15-350 keV) is 451.51 +- 585.16 sec (estimated error including systematics). The time-averaged spectrum from TTBD to TTBD sec is best fit by a power law with an exponential cutoff. This fit gives a photon index 1.12 +- 0.66, and Epeak of 19.1 +- 7.3 keV (chi squared 38.82 for 56 d.o.f.). For this model the total fluence in the 15-150 keV band is 7.5 +- 0.5 x 10^-06 erg/cm2 and the 1-sec peak flux measured from T-69.28 sec in the 15-150 keV band is 0.5 +- 0.2 ph/cm2/sec. A fit to a simple power law gives a photon index of 2.71 +- 0.10 (chi squared 64.64 for 57 d.o.f.). All the quoted errors are at the 90% confidence level. The results of the batgrbproduct analysis are available at http://gcn.gsfc.nasa.gov/notices_s/1047857/BA/