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The C$6.20 per share cash takeover via plan of arrangement was finalised on Friday, with South32 funding the acquisition via cash on hand.
It comes after 99.9% of Arizona shareholders voted in favour of the deal earlier this month.
Arizona was delisted from the Toronto Stock Exchange on Friday.
Arizona's advanced Hermosa project includes the high-grade Taylor resource of 101 million short tons at 10.4% zinc equivalent.
South32 CEO Graham Kerr said the acquisition of Arizona added one of the most exciting base metals projects in the industry to the company's portfolio.
"Our deep understanding of the high-grade Hermosa project and surrounding land package, together with our extensive experience at Cannington, positions us well to bring the project to development and deliver significant value to our shareholders," he said.
Work to enhance the design and development of the Hermosa Project will begin in the coming weeks.
The work will include optimising studies and the start of the exploration decline which will allow drilling to test for extensions and further enhance our understanding of the existing high-grade resource.
A January preliminary economic assessment outlined a US$519 million capital cost for a 29-year operation producing an average 946 million pounds of zinc and 9.5 million ounces of silver in the first five years at all-in sustaining costs of 61c per pound of zinc.
The PEA returned a post-tax net present value of $2 billion and a 1.6-year payback period.
South32 shares closed at A$3.63 on Friday, slightly below the $3.80 the stock was trading at when the deal was announced in June.