Using a sample of 771.6 x 10(6) Upsilon Upsilon(4S) decays collected by the Belle experiment at the KEKB e(+)e(-) collider, we observe, for the first time, the transition Upsilon(4S) -> eta h(b)(1P) with the branching fraction B[Upsilon(4S) -> eta h(b)(1P)] = (2.18 +/- 0.11 +/- 0.18) x 10(-3) and we measure the h(b)(1P) mass M-hb(1P) = (9899.3 +/- 0.4 +/- 1.0) MeV/c(2), corresponding to the hyperfine (HF) splitting Delta M-HF(1P) = (0.6 +/- 0.4 +/- 1.0) MeV/c(2). Using the transition h(b)(1P) -> gamma eta(b)(1S), we measure the eta(b)(1S) mass M-eta b(1S) = (9400.7 +/- 1.7 +/- 1.6) MeV/c(2), corresponding to Delta M-HF(1S) = (59.6 +/- 1.7 +/- 1.6) MeV/c(2), the eta(b)(1S) width Gamma(eta b)(1S) = (8(-5)(+6)+/- 5) MeV/c(2) and the branching fraction B[h(b)(1P) -> gamma eta(b)(1S)] = (56 +/- 8 +/- 4)%.