Amazon hasn't disclosed how many people have signed up for its Prime program, which offers free two-day shipping, a Netflix-like video service, and other perks for a $79 annual fee.

Now Morningstar analyst R.J. Hottovy has come out with a report that pegs membership at more than 10 million, and suggests that the program contributes about a third of Amazon's operating income.

Morningstar relied on a survey of Amazon shoppers conducted by market-research firm Consumer Intelligence Research Partners in November 2012.

After a slow start, Prime got a huge boost in when Amazon started including a 30-day trial membership with its Kindle Fire tablet in the fall of 2011. Trial programs for moms and students have also helped.

Prime members are great customers for Amazon. In 2012, they were about 4 percent of Amazon's 182 million active customer base. But they accounted for nearly 10 percent of purchases, because they spend more than twice as much as non-prime customers—$1,224 a year versus $505.

After estimating the cost of providing shipping and licensing videos and e-books versus the marginal profit , Hottovy found that Amazon made about $78 per Prime member in 2012.

While Amazon breaks even on the free shipping and other extras it provides Prime members, the subscription fee is almost pure profit, in other words.

By Hottovy's calculations, that could grow to 25 million by 2017. Depending on how much they spend, those members could contribute anywhere from $3.2 billion to $9.6 billion in incremental operating profits.

Of course, Amazon CEO Jeff Bezos has been known to plow its profits back into new products and lower prices, so shareholders may not see that cash hit the bottom line.

But Hottovy's research suggests that Amazon's grand strategy of linking its hardware business—Kindle Fire tablets—with its services—Prime, Instant Video, and its Kindle libraries—are paying off for its core e-commerce business.