This paper describes the calculation of the semileptonic K_l3 decay form
factor at order p^6 of chiral perturbation theory which is the next-to-leading
order correction to the well-known p^4 result achieved by Gasser and Leutwyler.
At order p^6 the chiral expansion contains 1- and 2-loop diagrams which are
discussed in detail. The irreducible 2-loop graphs of the sunset topology are
calculated numerically. In addition, the chiral Lagrangian L^6 produces direct
couplings with the W-bosons. Due to these unknown couplings, one can always add
linear terms in q^2 to the predictions of the form factor f_-(q^2). For the
form factor f_+(q^2), this ambiguity involves even quadratic terms. Making use
of the fact that the pion electromagnetic form factor involves the same q^4
counter term, the q^4-ambiguity can be resolved. Apart from the possibility of
adding an arbitrary linear term in q^2 our calculation shows that chiral
perturbation theory converges very well in this application, as the O(p^6)
corrections are small. Comparing the predictions of chiral perturbation theory
with the recent CPLEAR data, it is seen that the experimental form factor
f_+(q^2) is well described by a linear fit, but that the slope lambda_+ is
smaller by about 2 standard deviations than the O(p^4) prediction. The
unavoidable q^2 counter term of the O(p^6) corrections allows to bring the
predictions of chiral perturbation theory into perfect agreement with
experiment.Comment: 32 pages, 7 figure