The production of deuterons and antideuterons in the transverse momentum range 1.1 < p(T)< 4.3 GeV/c at midrapidity in Au+Au collisions at root s(NN) = 200 GeV has been studied by the PHENIX experiment at RHIC. A coalescence analysis, comparing the deuteron and antideuteron spectra with that of proton and antiproton, has been performed. The coalescence probability is equal for both deuterons and antideuterons and it increases as a function of p(T), which is consistent with an expanding collision zone. Comparing (anti)proton yields, (p) over bar /p = 0.73 +/- 0.01, with (anti)deuteron yields, (d) over bar /d = 0.47 +/- 0.03, we estimate that (n) over bar /n = 0.64 +/- 0.04. The nucleon phase space density is estimated from the coalescence measurement.