Without providing any actual figures Sprint says the Palm Pre smartphone, which launched early June, broke its own records for first day and weekend sales. Palm, on the other hand, only claims “sales have been strong and growing.” Indeed the device appears to be off to a good start, but with both companies mum on hard numbers, we have only the estimates of analysts with which to gauge its success.

The latest estimates come from Edward Snyder at Charter Equity Research, who says his sources in the manufacturing and retail channels indicate that Palm had sold 300,000 phones last month, and that the company has yet to catch up with demand. Palm is said to be producing 15,000 Pre units each day and plans to ship more than 1 million total units throughout its first quarter since launch.

That may not seem like much next to Apple’s million-in-a-weekend sales, but it’s still pretty impressive considering Palm shipped just 351,000 phones in the entire quarter prior to the Pre’s debut.