The Supreme Court decided the lethal injection case, Baze v. Rees today. The opinion is here. There is no majority, but I expect the plurality opinion by Chief Justice Roberts will be treated as the authoritative word. A couple of key passages:

Accordingly, we reject petitioners’ proposed “unnecessary risk” standard, as well as the dissent’s “untoward” risk variation. See post, at 2, 11 (opinion of Ginsburg, J.) [footnote omitted]

Instead, the proffered alternatives must effectively address a “substantial risk of serious harm.” Farmer, supra, at 842. To qualify, the alternative procedure must be feasible, readily implemented, and in fact significantly reduce a substantial risk of severe pain. If a State refuses to adopt such an alternative in the face of these documented advantages, without a legitimate penological justification for adhering to its current method of execution, then a State’s refusal to change its method can be viewed as “cruel and unusual” under the Eighth Amendment. [footnote omitted]

Slip op. at 11-12.

A stay of execution may not be granted on grounds such as those asserted here unless the condemned prisoner establishes that the State’s lethal injection protocol creates a demonstrated risk of severe pain. He must show that the risk is substantial when compared to the known and available alternatives. A State with a lethal injection protocol substantially similar to the protocol we uphold today would not create a risk that meets this standard.

Slip op. at 22 (emphasis added)