Help interpreting a binary logistic regression

Hi all,
I've a question about interpreting the results of a binary logistic regression. I'm using SPSS for my analysis.

I've one dependent which is a 0 or 1 categorical. I've then got a number of independent variables. Now, assuming that each of these are significant I read the Exp(B) value to show the effect of this variable. My question, or what I would like to check, is that the regression analysis takes account of the other variables.

So, if I make up a trivial example... the analysis could have independents including hair colour and height. If having brown hair had a significant odds ratio of above 1, could we say that this effect is independent of height? The case that I am concerned about is that perhaps people with brown hair tend to be taller and it's actually the height that is causing the effect on the dependent.

I hope this makes sense!