I'm going to take a different tack - but the answer is still "NO".
Let me tell a story first, and then get to the answer though.
Back in '07 I was working in San Fran, and I had two colleagues there, A and B. They were poorly named colleagues, but it is their experiences that are important here, not their parent's non-conformist view on naming.
A & B were both contractors initially, and both brought on full-time. B immediately took the perm position with minimal negotiating effort, and was consequently on a perm rate far lower than the contracting rate.
A did not. A, in fact, took 3 months negotiating with the company - they still paid her contracting rate because they needed the work done. After 3 months, A finally got across that she wanted the same wages as a contractor, only be a FTE. They agreed, I think there was some deal with guaranteed bonuses to help overcome the salary-band problems.
The point is this - if you don't need this job and are happy to look elsewhere, you are in an excellent position to negotiate. Why? Because you are taking a risk, in that they won't hire you and you will be out of a job. Risk, happily, is actually the argument proffered for why a company can pay you $10/hour for a job that nets them $40/hour - the risk the company takes (it might fail, for example) is worth the $30/hour they skim off you.
I would strongly advise you to take the course of action you are taking - but you have to know what the rate you want is. Do some homework - if it is worth the bump in pay you are implying, then do a lot of homework to work out what that rate is. Do not just shrug and say you want whatever the other company was paid, that makes you an amateur.
Remember that you are worth a large portion of the money you are asking for, as it is your market-rate. But perhaps do not gun for the entire rate - it is after all a negotiation, you are not trying to "prove a point", but create a win-win situation.