We present direct measurements at room temperature of the conductance of a point contact between a scanning tunneling microscope tip and Ni, Cu, and Pt surfaces. As the contact is stretched the conductance jumps in units of 2e2/h. Atomistic simulations of the stretch of the contact combined with calculations of the conductance using the Landauer formula show that the observed behavior is due to the quantization of the transverse electron motion in a contact which contains between one and ten atoms.