The first quasi-synoptic estimates of Arctic Ocean and sea ice net fluxes of volume, heat and freshwater are calculated by application of an inverse model to data around the ocean boundary. Hydrographic measurements from four gateways to the Arctic (Bering, Davis, and Fram Straits and the Barents Sea Opening) completely enclose the ocean, and were made within the same 32-day period in summer 2005. The inverse model is formulated as a set of full-depth and density-layer-specific volume and salinity transport conservation equations, with conservation constraints also applied to temperature, but only in non-outcropping layers. The model includes representations of Fram Strait sea ice export and of interior Arctic Ocean diapycnal fluxes. The results show that in summer 2005 the transport-weighted mean properties are, for water entering the Arctic: potential temperature 4.49°C, salinity 34.50 and potential density (σ0) 27.34 kg m−3; and for water leaving the Arctic, including sea ice: 0.25°C, 33.81, and 27.13 kg m−3, respectively. The net effect of the Arctic in summer is to freshen and cool the inflows by 0.69 in salinity and 4.23°C, respectively, and to decrease density by 0.21 kg m−3. The volume transport into the Arctic of waters above ∼1000 m depth is 9.2 Sv (1 Sv = 106 m3 s−1), and the export (similarly) is 9.3 Sv. The net oceanic and sea ice freshwater flux is 187 ± 48 mSv. The net heat flux (including sea ice) is 189 ± 37 TW, representing loss from the ocean to the atmosphere.