We present an interface conforming method for simulating two-dimensional and axisymmetric multiphase flows. In the proposed method, the interface is composed of straight segments which are part of mesh and move with the flow. This interface representation is an integral part of an Arbitrary Lagrangian-Eulerian (ALE) method on an moving adaptive unstructured mesh. Our principal aim is to develop an accurate and robust computational method for interfacial flows driven by strong surface tension and with weak viscous dissipation. We first construct a discrete solutions satisfying the Laplace law on a circular/spherical interfaces exactly, i.e., the balance between the surface tension and the pressure jump across an interface is achieved exactly. The accuracy and stability of these solutions are then investigated for a wide range of Ohnesorge numbers, $\backslash Oh$. The dimensionless amplitude of the spurious current is reduced to machine zero, i.e., on the order of $10-15$ for $\backslash Oh\ge 10-3$. Finally, the accuracy and capability of the proposed method are demonstrated through a series of benchmark tests with larger interface deformations. In particular, the method is validated with Prosperitti's analytic results of the bubble/drop oscillations and Peregrine's dripping faucet experiment, in which the values of $\backslash Oh$ are small.