GRB 210308***A***, Swift-BAT refined analysis S. D. Barthelmy (GSFC), H. A. Krimm (NSF), S. Laha (GSFC/UMBC), A. Y. Lien (U Tampa), C. B. Markwardt (GSFC), D. M. Palmer (LANL), T. Parsotan (GSFC/UMBC), T. Sakamoto (AGU), M. Stamatikos (OSU) (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 210308***A*** (trigger #1036227) (NAME, et al., GCN Circ. xxxx). The BAT ground-calculated position is RA, Dec = 67.081, 37.430 deg which is RA(J2000) = 04h 28m 19.6s Dec(J2000) = +37d 25' 46.7" with an uncertainty of 1.0 arcmin, (radius, sys+stat, 90% containment). The partial coding was 32%. [Description about the mask-weighted light curve] T90 (15-350 keV) is 6.2 +- 0.8 sec (estimated error including systematics). The time-averaged spectrum from T-0.0 to T+8.8 sec is best fit by a power law with an exponential cutoff. This fit gives a photon index 0.84 +- 0.28, and Epeak of 119.8 +- 41.7 keV (chi squared 55.46 for 56 d.o.f.). For this model the total fluence in the 15-150 keV band is 3.3 +- 0.1 x 10^-06 erg/cm2 and the 1-sec peak flux measured from T+2.12 sec in the 15-150 keV band is 13.6 +- 0.7 ph/cm2/sec. A fit to a simple power law gives a photon index of 1.39 +- 0.06 (chi squared 68.05 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/1036227/BA/