This paper concerns constructing wavelet bases on the unit interval, where a new boundary treatment is provided to overcome certain drawbacks of eariler constructions. Wavelet expansions on the unit interval usually suffer from artificial boundary effects and poor convergence at the boundaries. Many researchers have suggested a solution to the drawbacks. From a practical point of view, their solutions also have a common disadvantage. This paper provides a new solution using biorthogonality near the boundaries, that avoids the disadvantage while preserving their advantages.