  If the Higgs mass m_H is as low as suggested by present experimental
information, the Standard Model ground state might not be absolutely stable. We
present a detailed analysis of the lower bounds on m_H imposed by the
requirement that the electroweak vacuum be sufficiently long-lived. We perform
a complete one-loop calculation of the tunnelling probability at zero
temperature, and we improve it by means of two-loop renormalization-group
equations. We find that, for m_H=115 GeV, the Higgs potential develops an
instability below the Planck scale for m_t>(166\pm 2) GeV, but the electroweak
vacuum is sufficiently long-lived for m_t < (175\pm 2) \GeV.
