PROM in the case of the AGC was a magnetic core rope, not a packaged IC. While MIT designed and programmed the AGC, it was fabricated by Raytheon, who had lots of experience with hardened, flight-qualified computers for ICBMs. Apollo's guidance system was based largely on ICBM guidance components and methods. Raytheon workers -- typically women with lots of sewing experience -- threaded sense wires through (or not) cores to set (or clear) bits.
Fabricating a new rope was a huge deal, so there was quite a lot of attention paid to its correctness. While the attention paid to software correctness is not the same kind that is paid today, it had the effect of achieving considerably more rigor than is generally achieved today.
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