We have determined the magnetic phase diagram of the quasi-one-dimensional $S=$ 1 Heisenberg antiferromagnet $Ni (C_5 H_{14} N_2)_2 N_3 (PF_6)$ by specific heat measurements to 150 mK in temperature and 32 T in magnetic field. When field is applied along the spin-chain direction, a new phase appears at $H_{c2}approx 14$ T. For the previously known phases of field-induced order, accurate determination is made of the power-law exponents of the ordering temperature near the zero-temperature critical field $H_c$, owing to the four-fold improvement of the minimum temperature over the previous work. The results are compared with the predictions based on the Bose-Einstein condensation of triplet excitations. Substituting deuterium for hydrogen is found to slightly reduce the interchain exchange.
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