How can I get my gas oven to light?

I have an old O'Keefe & Merritt stove (similar to those shown at
http://www.antiquegasstoves.com/pages/okeefe.html). The oven pilot is
burning continuously.

In order to light the oven, instructions on the door tell me I'm
supposed to hold the red button on a safety valve under the stove top
for 45 seconds. While I hold the red button and turn the oven's gas
control on the front of the stove, gas flows and the oven lights and
heats up. But as soon as I release the button the gas stops flowing
and the oven goes off. Even if I make a point of holding in the button
for longer than 45 seconds.

What isn't working? I think the oven has a safety mechanism that makes
sure the pilot is on before it will let gas flow. It needs to detect
heat from the pilot. My red button is overriding that. In that case, I
think the safety mechanism heat isn't operating right. It doesn't
figure out that the pilot is on OK. But by inspection, I don't see
such a mechanism. I'm not sure what to do next. What should I look
for.

Any suggestions?