Without getting nitpicky to the point where it's impossible to say anything at all without complaining it's not a perfectly complete and true statement, their statement still seems correct to me. Especially given that this is a statement for non-statisticians and they can't exactly go on a 5 minute tangent to explain exactly how p-values work.
Perhaps it would have been more accurate if they phrased it: "which means a chance of less than one in a billion of seeing a correlation this strong if it's due to chance", but I think few readers would even notice the difference.
>P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.
It was very informative for me. Worth reading.