Summary: We prove the following unique continuation property. Let \(u\) be a solution of a second order linear parabolic equation and \(S\) a segment parallel to the \(t\)-axis. If \(u\) has a zero of order faster than any nonconstant and time independent polynomial at each point of \(S\), then \(u\) vanishes at each point \((x,t')\) such that the plane \(t=t'\) has a nonempty intersection with \(S\).