= Solution
Assume the parents are unaffected, the disease has complete <penetrance> with a single recessive causal <allele>, and there are no phenocopies, new disease <mutations> or alternative causal loci. An affected first child has <genotype> $aa$, so each unaffected parent must be a <heterozygote> $Aa$. Conditional on these parental <genotypes>, <Mendelian segregation> in the second child's paternal and maternal <meioses> is independent of the transmissions to the first child. Therefore
$$
\boxed{P(\text{second affected}\mid\text{first affected, parents unaffected})=\frac12\cdot\frac12=\frac14.}
$$
\b[This is the same sibling recurrence risk as for unrelated, unaffected parents under the same recessive model.] Parental kinship increases the <prior> <probability> that both parents carry the same rare disease <allele>; after both are identified as carriers, it does not alter their Mendelian transmission <probabilities>. The factor $63.4$ from the preceding part is therefore not an additional multiplier on the <sibling recurrence risk>.
If unaffected parents are not explicitly required, the rare-disease assumption is an approximation supporting the usual carrier-by-carrier calculation, rather than a logically exact consequence of an affected child alone. Under complete <penetrance>, the compatible parental matings are $Aa\times Aa$, $Aa\times aa$ and $aa\times aa$, with second-child risks $1/4,1/2,1$ respectively. Let their <prior> <probabilities> be $w_{11},w_{12},w_{22}$, including both orders in $w_{12}$. Conditioning on the affected first child gives
$$
P(\text{second affected}\mid\text{first affected})=\frac{w_{11}/16+w_{12}/4+w_{22}}{w_{11}/4+w_{12}/2+w_{22}}.
$$
Without negligible or excluded affected-parent matings, this mixture need not equal $1/4$ and may depend on consanguinity. Likewise incomplete <penetrance> or disease heterogeneity invalidates the assertion that an affected child of unaffected parents necessarily identifies an $Aa\times Aa$ mating. The stated $1/4$ answer uses the unaffected-parent, fully penetrant single-locus assumptions.
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