We prove that all subgroups
H
H
of a free product
G
G
of two groups
A
,
B
A,B
with an amalgamated subgroup
U
U
are obtained by two constructions from the intersection of
H
H
and certain conjugates of
A
,
B
A,B
, and
U
U
. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-Neumann-Neumann group. The particular conjugates of
A
,
B
A,B
, and
U
U
involved are given by double coset representatives in a compatible regular extended Schreier system for
G
G
modulo
H
H
. The structure of subgroups indecomposable with respect to amalgamated product, and of subgroups satisfying a nontrivial law is specified. Let
A
A
and
B
B
have the property
P
P
and
U
U
have the property
Q
Q
. Then it is proved that
G
G
has the property
P
P
in the following cases:
P
P
means every f.g. (finitely generated) subgroup is finitely presented, and
Q
Q
means every subgroup is f.g.;
P
P
means the intersection of two f.g. subgroups is f.g., and
Q
Q
means finite;
P
P
means locally indicable, and
Q
Q
means cyclic. It is also proved that if
N
N
is a f.g. normal subgroup of
G
G
not contained in
U
U
, then
N
U
NU
has finite index in
G
G
.