This paper describes the calculation of the semileptonic Kl3 decay form factors at order p6 of chiral perturbation theory, which is the next-to-leading order correction to the well-known p4 result achieved by Gasser and Leutwyler. At order p6 the chiral expansion contains one- and two-loop diagrams which are discussed in detail. The irreducible two-loop graphs of the sunset topology are calculated numerically. In addition, the chiral Lagrangian \documentclass[12pt]{minimal}
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\begin{document}${\cal L}^{(6)}$\end{document} produces direct couplings with the W bosons. Due to these unknown couplings, one can always add linear terms in q2 to the predictions of the form factor f-(q2). For the form factor f+(q2), this ambiguity involves even quadratic terms. Making use of the fact that the pion electromagnetic form factor involves the same q4 counterterm, the q4 ambiguity can be resolved. Apart from the possibility of adding an arbitrary linear term in q2 our calculation shows that chiral perturbation theory converges very well in this application, as the \documentclass[12pt]{minimal}
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\begin{document}${\cal O}(p^{6})$\end{document} corrections are small. Comparing the predictions of chiral perturbation theory with the recent CPLEAR data, it is seen that the experimental form factor f+(q2) is well described by a linear fit, but that the slope \documentclass[12pt]{minimal}
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\begin{document}$ \lambda _{+}$\end{document} is smaller by about 2 standard deviations than the \documentclass[12pt]{minimal}
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\begin{document}${\cal O} (p^{4})$\end{document} prediction. The unavoidable q2 counterterm of the \documentclass[12pt]{minimal}
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\begin{document}${\cal O}(p^{6})$\end{document} corrections allows one to bring the predictions of chiral perturbation theory into perfect agreement with experiment.
