  We present a detailed study of D-branes in the axially gauged SL(2,R)/U(1)
coset conformal field theory for integer level k. Our analysis is based on the
modular bootstrap approach and utilizes the extended SL(2,R)/U(1) characters
and the embedding of the parafermionic coset algebra in the N=2 superconformal
algebra. We propose three basic classes of boundary states corresponding to
D0-, D1- and D2-branes. We verify that these boundary states satisfy the Cardy
consistency conditions and discuss their physical properties. The D0- and
D1-branes agree with those found in earlier work by Ribault and Schomerus using
different methods (descent from the Euclidean AdS3 model). The D2-branes are
new. They are not, in general, space-filling but extend from the asymptotic
circle at infinity up to a circular boundary at some distance from the tip of
the cigar.
