This is the draft order for the 1st round of June's draft. Changes to the 1st round order due to FA signings are in italics. The supplemental 1st round is next, followed by a list of changes to the 2nd round, the supplemental 2nd round, and lists of changes to the 3rd and 4th rounds. Jeff Weaver, a Type A, is the only FA left who could affect the draft order.
1st-round
1. Royals (56-106)
2. Rockies (67-95)
3. Devil Rays (67-95)
4. Pirates (67-95)
5. Mariners (69-93)
6. Tigers (71-91)
7. Dodgers (71-91)
8. Reds (73-89)
9. Orioles (74-88)
10. Giants (75-87)
11. Diamondbacks (77-85)
12. Rangers (79-83)
13. Cubs (79-83)
14. Blue Jays (80-82)
15. Nationals (81-81)
16. Brewers (81-81)
17. Padres (82-80)
18. Phillies (88-74) - (from Mets for Type A Billy Wagner)
19. Marlins (83-79)
20. Twins (83-79)
21. Yankees (95-67) - (from Phillies for Type A Tom Gordon)
22. Nationals (81-81) - (from Athletics for Type B Esteban Loaiza)
23. Astros (89-73)
24. Braves (90-72)
25. Angels (95-67) - (from Guardians for Type B Paul Byrd)
26. Nationals (81-81) - (from Angels for Type B Hector Carrasco)
27. Red Sox (95-67)
28. Red Sox (95-67) - (from Yankees for Type A Johnny Damon)
29. White Sox (99-63)
30. Cardinals (100-62)
Supplemental 1st-round
31. Orioles (for Type A B.J. Ryan)
32. Giants (for Type A Scott Eyre)
33. Diamondbacks (for Type A Tim Worrell)
34. Padres (for Type A Ramon Hernandez)
35. Marlins (for Type A A.J. Burnett)
36. Phillies (for Wagner)
37. Braves (for Type A Kyle Farnsworth)
38. Guardians (for Type A Bob Howry)
39. Red Sox (for Damon)
40. Yankees (for Gordon)
41. Cardinals (for Type A Matt Morris)
42. Braves (for Type A Rafael Furcal)
43. Red Sox (for Type A Bill Mueller)
2nd-round
50. Braves (from Dodgers for Furcal)
52. Padres (from Orioles for Hernandez)
53. Cardinals (from Giants for Morris)
55. Guardians (from Rangers for Type B Kevin Millwood)
56. Guardians (from Cubs for Howry)
57. Orioles (from Blue Jays for Ryan)
71. Braves (from Yankees for Farnsworth)
Supplemental 2nd-round
74. Guardians (for Type C Scott Elarton)
75. Cardinals (for Type C Abraham Nunez)
3rd-round
82. Red Sox (from Dodgers for Mueller)
85. Diamondbacks (from Giants for Worrell)
88. Giants (from Cubs for Eyre)
89. Marlins (from Blue Jays for Burnett)
4th-round
118. Twins (from Cubs for Type B Jacque Jones)
The Red Sox have pick #'s 27, 28, 39, 43, 60, 82, and 102 of the first three rounds.
Source for FA related changes - BaseballAmerica.com