Unique solvability of the one-phase Stefan problem with a small multiplier ε at the time derivative in the equation is proved on a certain time interval independent of ε for ε (0, ε0). The solution to the Stefan problem is compared with the solution to the Hele-Show problem, which describes the process of melting materials with zero specific heat ε and can be regarded as a quasistationary approximation for the Stefan problem. It is shown that the difference of the solutions has order script O sign (ε) + script O sign (e-ct/ε). This provides a justification of the quasistationary approximation. Bibliography: 23 titles. © 2008 Springer Science+Business Media, Inc.