The 1st Minnesota Infantry served as a member of Harrow’s Brigade in Gibbon’s Division of the Second Corps, Army of the Potomac, a Fighting 300 Regiment. The unit was commanded by Colonel William Colvill (1830-1905), a lawyer and newspaper editor in Red Wing. Colvill was wounded at Glendale and again on July 2 while here @ Gettysburg. Colvill who was hit three times and severally wounded. He was shot in the shoulder and the ankle. One bullet entered the top right shoulder and tore across his back, clipping off a part of his vertebra and lodging under his left scapula. Both wounds would force Colvill to use a cane the rest of his life. In his post-war life he was a state legislature and state attorney general. Under Colvill's command 420 men were engaged at Gettysburg and among them, 50 were killed and 173 were wounded and 1 went missing. From my experience, this regiment suffered HUGE losses compared to their counterparts who also fought here at Gettysburg. The 83.1 percent casualty rate stands to this day as the largest loss by any surviving military unit in U.S. history during a single day's engagement. The unit's flag is now in the Minnesota Capitol's rotunda. In 1905, Colvill traveled to the Soldiers Home in Minneapolis to attend a reunion of the veterans of the First Minnesota. While there, he died in his sleep on June 13. He is buried in the Cannon Falls Community Cemetery in Cannon Falls, Minnesota. In 1928, President Calvin Coolidge was present for the dedication of a statue that was placed next to his grave.
As already reported, Colonel William Colville led the men into action and was wounded during the July 2 charge. Captain Nathan Messick took over command only to be killed the next day during the repulse of Pickett's Charge. He was briefly followed by Captain Wilson B. Farrell, who was also killed, and finally by Captain Henry C. Coates. The men of the 1st Minnesota are most remembered for their actions on July 2, 1863, during the second day's fighting at Gettysburg, where the regiment prevented the Confederates from pushing the Federals off of Cemetery Ridge, a position that was to be crucial in the battle.
The primary 
1st Minnesota Monument can be found at the intersection of South Hancock Avenue and Humphreys Avenue (the two roads make a "y"), on the east or left side of the road when traveling north on Hancock Avenue.
This memorial urn is located inside the National Cemetery. One would go in through the Baltimore Pike entrance and stay to the right, following the path, going past the New York monument. Soon you will see sections of graves for individual states. This memorial is located in front of Minnesota section of the cemetery. It is 131 feet southwest of the National monument and 283 feet east of the Taneytown Road wall. As a point of reference, the Brevet Major General Charles H. T. Collis Memorial (MN406) is 137 feet northwest of the Minnesota urn. 
Parking is available in front of the cemetery at metered spots along the curb. There is also a lot located to the left of East Cemetery Hill (if facing it) in a tourist-tour business that always has space. I visited this site on Monday, July 1, 2013 on the 150th anniversary of the Battle of Gettysburg @ 11:50 AM, EDT & @ an altitude of 619 feet, ASL. As always, I used my trusty and oft abused Canon PowerShot 14.1 Megapixel, SX210 IS digital camera for the photos.
The Draw the Sword site helped out by the NPS narrative offers the following description: Urn located in the National Cemetery, Minnesota plot. Placed in 1867, making it one of the battlefield’s oldest monuments. The state of Minnesota erected this white marble vase-on-stand at the head of its burial section. It is a stepped shaft topped with a sculpted, reverse fluted marble urn that is set on a 2.10 foot square base. The overall height is 5.2 foot. Although I was unable to uncover any information as to the artisans employed to produce the urn, I did discover the name J. Cannan written on the front, lower right hand of the base. Inscriptions on the square stand include words from the principal addresses during the cemetery’s 1863 dedication.
While the Soldiers National Monument was being constructed and finished, in 1867, the First Minnesota Regiment, which had been virtually annihilated on the 2nd day of battle at Gettysburg, erected a memorial urn in the National Cemetery at the site of the Minnesota plot. The white marble vase-on-stand is at the head of its burial section. The monument was built in 1867 and formally dedicated in 1869 by the Surviving Members of the First Minnesota Infantry. The monument is composed of marble (I think). There are brief on all four sides, some of which contain attributable quote, all of which read:
(Front):
First Minnesota Volunteers
(Left):
"All time is the millenium of their glory."
(Right):
These dead shall not have died in vain.
(Back):The surviving members
First Regiment
Minnesota Infanty.
To the memory
of their late associates
who
"died on the field of honor"
at Gettysburg
July 1863.
The 1st Minnesota Infantry Memorial is a contributing feature to the Gettysburg National Military Park Historic District which is nationally significant under NR Criteria A, B, C & D. Areas of Significance: Military, Politics/Government, Landscape Architecture, Conservation, Archeology-Historic. Period of Significance: 1863-1938. The original National Register Nomination was approved by the Keeper March 19, 1975. An update to this nomination was approved by the Keeper on January 23, 2004. The monument is identified as structure number MN639.
From the Nomination Form:
1 of 3 Civil War Monuments to Minnesota. Memorial to Minnesota soldiers interred in National Cemetery. It was the first built memorial of any kind either in the national cemetery or on the battlefield. Memorial located in front of Minnesota section of cemetery.
Short Physical Description:
Mn base 2'10"sq, stepped shaft w/ main part inscribed all 4 sides. Topped w/ sculpted, reverse fluted marble urn 5'4" in diameter. All 5'2"H. Inscriptions on stand include words from the principal addresses during the cemetery's 1863 dedication.
Long Physical Description:
The state of Minnesota erected this white marble vase-on-stand at the head of its burial section. It is a stepped shaft topped with a sculpted, reverse fluted marble urn that is set on a 2.10 foot square base. The overall height is 5.2 foot. Inscriptions on the square stand include words from the principal addresses during the cemetery’s 1863 dedication. It was the first built memorial of any kind either in the national cemetery or on the battlefield.
My Sources
1.
NRHP Nomination Form
2.
Stone Sentinels
3.
Virtual Gettysburg
4.
Draw the Sword
5.
Historical Marker Database