I have an annuity that is held within an IRA. I plan to exercise the annuity, and I assume this will take the resultant income stream (life/10 year certain) outside of the IRA.

I assume that there are no taxes on the act of exercising the annuity and taking the income stream out of the IRA, but that taxes will be paid each year of the income stream.

Is this correct?

You are correct - the annuity effectively remains inside the IRA (or it may roll directly to a new IRA account), and so taxes are due only as you take distributions.

One comment however - I don't mean to preach, but whomever advised you to invest in an annuity inside your IRA did you a disservice. Annuities are already tax-deferred , so you essentially wasted the opportunity for the additional tax-deferral that is available to you in an IRA. In other words, you most likely would have been better off using your IRA contributions to invest in things that are normally not tax-deferred (mutual funds, stocks, bonds, etc) and then you could get more tax-defferal by purchasing the annuity in a separate account which is not part of your IRA. Given the fees that most annuities charge and the restrictions they impose, the general order of priority would be to invest first in an IRA (or even better - a 401(k) if you have one at work), and if you have additional cash left over after that then consider investing in an annuity. Another approach is to invest in an IRA until you reach retirement, then roll it to an annuity at that time so that you have a secure, steady income stream for life.