We propose a strategy to build portfolios of sovereign securities with progressively declining carbon footprints. Passive investors could use it as a new Paris-consistent benchmark to construct a "net zero" (NZ) portfolio while tracking closely the risk-adjusted returns of a business-as-usual (BAU) benchmark. Our strategy rewards sovereign issuers that have made stronger efforts in reducing carbon intensity, measured by total domestic emissions per capita. The NZ portfolio would have reduced carbon intensity by 41% between 2014 and 2019, by assigning higher weights to countries that have had lower carbon emissions. Among advanced economies, rebalancing leads to raising shares of France, Italy and Spain in the portfolio at the expense of the United States. And among emerging market economies, this leads to higher shares for Chile, the Philippines and Romania at the expense of China. Importantly, the NZ portfolio retains the same creditworthiness as the BAU benchmark without entailing materially higher foreign exchange risks.