To study contaminant transport in groundwater, an essential requirement is robust and accurate estimation of the transport parameters such as dispersion coefficient. The commonly used inverse error function method (IEFM) may cause unacceptable errors in dispersion coefficient estimation using the breakthrough curves (BTCs) data. We prove that the random error in the measured concentrations, which might be described by a normal distribution, would no longer follow the normal distribution after the IEFM transformation. In this study, we proposed a new method using the weighted least squares method (WLSM) to estimate the dispersion coefficient and velocity of groundwater. The weights were calculated based on the slope of the observed BTCs. We tested the new method against other methods such as genetic algorithm and CXTFIT program and found great agreement. This new method acknowledged different characteristics of solute transport at early, intermediate, and late time stages and divided BTCs into three sections for analysis. The developed method was applied to interpret three column tracer experiments by introducing continuous, constant-concentration of sodium chloride (NaCl) into columns filled with sand, gravel, and sand-gravel media. This study showed that IEFM performed well only when the observed data points were located in the linear (intermediate time) section of BTCs; it performed poorly when data points were in the early and late time stages. The new WLSM method, however, performed well for data points scattering over the entire BTCs and appeared promising in parameter estimation for solute transport in a column.