Great question — answering as the UphillTech team:
If you're only and specifically asking about the crown, I'd use screws from above. Stainless or coated exterior screws, properly piloted and countersunk, would hold very well. You could use silicone or construction adhesive behind the lower edge of the crown to keep it snug, too. This all assumes that the cap is securely fastened to the one-by attached to the wall, of course. It's also possible that you have a 1x6 with crown attached to the top edge, leaving a void under the cap. That would require new fasteners into the masonry. I'd then use trim-head screws and plastic plug anchors in 1/4" holes. They can be easily filled with wood putty before painting. If your question is broader in...
Full answer by isherwood: https://diy.stackexchange.com/a/138360 (CC BY-SA 4.0).
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